Algebraic topology bridges two distinct mathematical worlds: the continuous realm of topological spaces and the discrete structures of algebra. Rather than studying topological properties directly through geometric intuition, this course develops systematic methods to encode topological information into algebraic objects—primarily groups and their homology—whose properties are often easier to compute and compare. By translating shape-theoretic questions into algebraic equations, we gain powerful tools to distinguish spaces, classify maps, and understand how spaces decompose into simpler pieces. The course centers on three interrelated algebraic constructions. The fundamental group captures how loops can be continuously deformed, serving as an invariant that detects when spaces have "holes" in their first dimension. Covering spaces provide a complementary perspective: they allow us to understand global properties of a space by studying its locally uniform sheets. Simplicial homology measures holes systematically in all dimensions, providing a dimension-by-dimension accounting of topological structure that is more computable than the fundamental group alone. The chapters build progressively from concrete foundations to abstract unification. Early chapters establish the foundational definitions and develop homotopy theory, which quantifies when two continuous maps should be considered "the same" from a topological standpoint. The treatment of the fundamental group and covering spaces introduces key computational techniques, while the Seifert–van Kampen theorem demonstrates how these invariants behave under decomposition. The final chapters transition to simplicial methods: simplicial complexes provide a combinatorial framework for approximating arbitrary spaces, and simplicial homology translates this framework into explicit algebraic computations. Together, these structures form a toolkit for solving problems that appear intractable when approached through topology alone. # 1. Introduction Algebraic topology begins with a deceptively simple question: when are two topological spaces the same? If they are homeomorphic, one can exhibit an explicit homeomorphism. But proving that two spaces are *not* homeomorphic is a harder problem — you must rule out every possible continuous bijection. This chapter motivates the central strategy of the course: translate topological non-existence problems into algebraic ones, where we have far more powerful tools for proving impossibility. Along the way we set up the basic language and building blocks — maps, the gluing lemma, and cell complexes — that will be used throughout. ## The Central Strategy: Algebra from Topology In topology, a typical problem is: given spaces $X$ and $Y$, is $X \cong Y$, i.e.\ are they homeomorphic? Exhibiting a homeomorphism settles the matter in one direction. The difficulty lies in the negative direction: how do we prove that no homeomorphism exists? [motivation] ### The Problem with Pure Topology Consider $\mathbb{R}^m$ and $\mathbb{R}^n$ for $m \neq n$. Intuitively they are not homeomorphic — they have different dimensions — but "different dimensions" is not itself a proof. To make this rigorous we need a topological invariant: some quantity attached to a space that is preserved under homeomorphism, so that if the two spaces have different values of the invariant, they cannot be homeomorphic. Pure topology offers some invariants (connectedness, compactness, ...) but these are too coarse to distinguish $\mathbb{R}^m$ from $\mathbb{R}^n$. Both are connected, locally compact, and paracompact. We need something finer. ### Translation to Algebra The idea of algebraic topology is to assign algebraic objects — groups, rings, modules — to topological spaces in a way that is functorial: a continuous map $f: X \to Y$ induces a homomorphism between the associated algebraic objects. This means that: - A homeomorphism $f: X \to Y$ induces an isomorphism between the algebraic objects. - Therefore, if the algebraic objects of $X$ and $Y$ are not isomorphic, the spaces cannot be homeomorphic. We have traded a hard topological problem (classifying homeomorphisms) for an algebraic problem (classifying group isomorphisms). The power of this trade is that algebra gives us many concrete tools — abelianisation, rank, torsion — that let us distinguish groups rapidly. ### An Illustrative Reduction The course will reduce the question "is $\mathbb{R}^m \cong \mathbb{R}^n$ for $m \neq n$?" to the question "is $\mathbb{Z} \cong \{e\}$?" The latter is immediate: $\mathbb{Z}$ has infinitely many elements, while $\{e\}$ has one. So $\mathbb{Z} \not\cong \{e\}$ as groups, and therefore $\mathbb{R}^m \not\cong \mathbb{R}^n$. [/motivation] The strategy also handles more surprising results. Let $D^n$ denote the closed $n$-dimensional unit disk, and $S^{n-1}$ the $(n-1)$-dimensional unit sphere (the boundary of $D^n$). A natural question is: can we continuously retract $D^n$ onto its boundary $S^{n-1}$? That is, does there exist a continuous map $F: D^n \to S^{n-1}$ such that the composite \begin{align*} S^{n-1} \hookrightarrow D^n \xrightarrow{F} S^{n-1} \end{align*} is the identity? Geometrically, this asks whether one can continuously "push" the entire disk onto its boundary while leaving the boundary fixed. Algebraic topology translates this into: does there exist a group homomorphism $F: \{0\} \to \mathbb{Z}$ such that the composite \begin{align*} \mathbb{Z} \hookrightarrow \{0\} \xrightarrow{F} \mathbb{Z} \end{align*} is the identity? Since the only homomorphism out of the trivial group $\{0\}$ sends everything to $0$, the composite sends $1 \in \mathbb{Z}$ to $0$, which is not the identity on $\mathbb{Z}$. So no such $F$ exists — and therefore the continuous retraction cannot exist either. This is the template: a hard topological non-existence statement becomes an easy algebraic non-existence statement. [remark: Applications Beyond Homeomorphism] The methods of algebraic topology extend well beyond homeomorphism classification. The same machinery proves Brouwer's fixed point theorem — every continuous map $D^2 \to D^2$ has a fixed point — and can be used to prove the fundamental theorem of algebra. These are theorems about solutions of equations, not about the shape of spaces, yet algebraic topology provides some of the cleanest proofs. [/remark] ## Conventions and the Gluing Lemma Suppose we want to define a function $f: \mathbb{R}^2 \to \mathbb{R}$ by one formula on the closed upper half-plane $\{y \geq 0\}$ and a different formula on the closed lower half-plane $\{y \leq 0\}$ — say $f(x, y) = x^2 + y$ for $y \geq 0$ and $f(x, y) = x^2 - y$ for $y \leq 0$. The two formulas agree on the real axis $\{y = 0\}$, so $f$ is well-defined, and each formula is continuous on its own half-plane. But is the resulting patchwork $f$ continuous as a function on all of $\mathbb{R}^2$? That is the fundamental question about piecewise definitions: when does gluing continuous pieces along their overlap produce a continuous whole? The answer is not automatic — it depends on whether the pieces are closed (or both open). The gluing lemma makes the answer precise and will be invoked constantly when constructing maps between cell complexes. [definition: Map] In this course, the word *map* always means *continuous map*. We are doing topology, and non-continuous functions will never be considered. All morphisms between topological spaces are continuous. [/definition] We will repeatedly define maps out of quotient spaces and cell complexes by piecing together maps defined on individual closed cells or closed pieces. Whenever we do this, we need to know that the resulting patchwork is continuous — and, conversely, that verifying continuity can be reduced to checking it on each piece separately. The following lemma is the tool that licenses every such construction in the sequel. [quotetheorem:1871] [citeproof:1871] The hypothesis that $C$ and $K$ are both *closed* is essential. To see why, split $\mathbb{R}$ as $(-\infty, 0]$ (closed) and $(0, \infty)$ (open, not closed), and define $f$ to be $-1$ on the first piece and $+1$ on the second. Each restriction is continuous, but the resulting function (essentially the sign function) is discontinuous at $0$: the point $0$ is in the closed piece, but $f$ jumps at $0$. The gluing lemma also holds with "closed" replaced by "open" throughout (by the same argument, now using preimages of open sets); and by induction it extends to any *finite* closed cover. It is equally important to note what the lemma does **not** say: finiteness of the cover cannot be dropped. In any $T_1$ space every singleton $\{x\}$ is closed, so $X = \bigcup_{x \in X} \{x\}$ is a closed cover and every restriction $f|_{\{x\}}$ is trivially continuous — if the lemma extended to infinite closed covers, every function would be continuous, which is absurd. The finite closed cover hypothesis is genuinely needed. Another technical tool we will need, especially when working with covers of compact metric spaces, is the existence of a uniform scale below which every small ball fits inside some element of the cover. [quotetheorem:952] [citeproof:952] Compactness is indispensable here. On the open interval $(0,1)$ with the cover $\{(1/(n+1), 1/n)\}_{n \geq 1}$, no Lebesgue number exists: points near $0$ are not contained in any single element of the cover by a ball of fixed radius. The lemma is also intrinsically metric: it uses the uniform structure of a metric space to pass from pointwise containment (each $x$ lies in some $U_\alpha$) to a uniform lower bound $\delta$. What the lemma does **not** assert is that $\delta$ depends only on the space $X$: the Lebesgue number is a property of the cover, not of $X$ alone. Different covers of the same compact metric space generally have different Lebesgue numbers, and refining a cover typically shrinks $\delta$. The lemma guarantees existence of $\delta > 0$ for each given cover, not a universal constant. The Lebesgue number lemma will appear later when we need to subdivide paths and homotopies into pieces that each lie within a given open set — a technique central to the proof of the path-lifting property for covering spaces. ## Cell Complexes Consider the *Hawaiian earring*: the subspace of $\mathbb{R}^2$ consisting of circles of radius $1/k$ centred at $(1/k, 0)$ for $k = 1, 2, 3, \ldots$, all tangent to the origin. It is compact, path-connected, and a subspace of the plane — yet its fundamental group is uncountable and notoriously hard to describe. Other familiar-looking spaces are worse: the long line is path-connected and locally Euclidean but not metrizable, and arbitrary topological spaces can fail to be Hausdorff, first countable, or even $T_1$. For such spaces, the algebraic invariants we are about to construct either fail to exist or become intractable to compute. To keep the theory workable — and, crucially, to have spaces whose invariants can actually be calculated — algebraic topology restricts attention to *cell complexes*, spaces built inductively by gluing disks along their boundaries. This restriction rules out the Hawaiian earring and the long line but retains every space we care about: spheres, tori, projective spaces, surfaces, Lie groups, and more. The basic operation is attaching a single disk along its boundary. [definition: Cell Attachment] Let $X$ be a topological space and $f: S^{n-1} \to X$ a continuous map. The space obtained by *attaching an $n$-cell to $X$ along $f$* is the quotient \begin{align*} X \cup_f D^n \;=\; (X \amalg D^n) \,/\, {\sim}, \end{align*} where $\sim$ is the equivalence relation generated by $x \sim f(x)$ for all $x \in S^{n-1} \subset D^n$. Here $\amalg$ denotes disjoint union. [/definition] Intuitively: the map $f: S^{n-1} \to X$ picks out a copy of the sphere inside $X$. We then glue the disk $D^n$ onto $X$ by identifying each boundary point $x \in S^{n-1} = \partial D^n$ with its image $f(x) \in X$. The interior of the disk becomes a new open $n$-cell. [illustration:cell-attachment-general] Attaching a single cell is the atomic operation. To build interesting spaces we iterate it: start from a finite set of points, attach $1$-cells to produce a graph, then attach $2$-cells to fill in faces, then $3$-cells, and so on. Each stage takes the previous skeleton and glues on a collection of cells of the next dimension simultaneously via attaching maps. The resulting inductive procedure is the definition of a cell complex. [definition: Cell Complex] A (finite) *cell complex* (also called a CW complex in the finite case) is a space $X$ constructed by the following inductive procedure: 1. Start with a finite discrete set $X^{(0)}$, called the *0-skeleton*. Its points are the *0-cells*. 2. Given the $(n-1)$-skeleton $X^{(n-1)}$, form the *$n$-skeleton* $X^{(n)}$ by choosing a finite collection of maps $\{f_\alpha: S^{n-1} \to X^{(n-1)}\}$ and setting \begin{align*} X^{(n)} \;=\; \left(X^{(n-1)} \amalg \coprod_\alpha D^n_\alpha\right) \Big/ \{x \sim f_\alpha(x)\}. \end{align*} Each open disk $\operatorname{int}(D^n_\alpha)$ maps homeomorphically onto an open subset of $X^{(n)}$, called an *$n$-cell*. 3. Stop at some $X = X^{(k)}$. The smallest such $k$ is the *dimension* of $X$. For infinite cell complexes, drop the finiteness conditions and allow infinitely many cells at each stage; the topology on the resulting space is then defined by declaring a set to be closed iff its intersection with each $X^{(n)}$ is closed. [/definition] [illustration:cell-complex-skeleton-construction] To see the definition in action — and to appreciate how economical cell decompositions can be — it helps to start with the most important family of examples. The spheres admit a particularly minimal cell structure: each $S^n$ is built from just one $0$-cell and one $n$-cell. The example below makes the attaching map and the resulting quotient explicit, and will be the prototype for more elaborate cell decompositions later. [example: The Spheres as Cell Complexes] The $n$-sphere $S^n$ has a minimal cell complex structure with exactly two cells: a single 0-cell (a point $p$) and a single $n$-cell. The attaching map $f: S^{n-1} \to X^{(0)} = \{p\}$ is the unique map, which sends all of $S^{n-1}$ to $p$. The resulting quotient $\{p\} \cup_f D^n = D^n / S^{n-1}$ is homeomorphic to $S^n$: the explicit homeomorphism sends the collapsed boundary class (the single point $[S^{n-1}]$ in the quotient) to the north pole of $S^n$, and maps each interior point of $D^n$ radially outward onto the corresponding point of $S^n \setminus \{\text{north pole}\}$ via a radial extension of stereographic projection from the south pole. Collapsing the boundary of the disk to a point thus produces a sphere. For $n = 1$: we start with one point $p$, attach a 1-cell (an interval $[0,1]$) by sending both endpoints to $p$. The result is a circle $S^1$. For $n = 2$: we attach a 2-disk to $p$ by collapsing its boundary circle to $p$, giving $S^2$. [/example] The sphere example shows that cell complex structures can be remarkably economical — even a high-dimensional sphere needs only two cells. But the existence of such structures is not guaranteed for every space; the next example shows that the cell-complex condition has genuine content. [example: A Non-Example] Not every reasonable-looking space is a cell complex. Consider the subspace of $\mathbb{R}^2$ consisting of the circles of radius $1/2^k$ and centre $(0, 1/2^k)$ for $k = 1, 2, 3, \ldots$. These circles are all tangent to the origin, and they accumulate there. This space — sometimes called the Hawaiian earring — is not a cell complex. In a cell complex, every compact subset meets only finitely many cells; but in the Hawaiian earring, every neighbourhood of the origin meets infinitely many circles, so no cell decomposition is possible. [/example] The Hawaiian earring is a cautionary example, but in practice it illustrates a general principle: restricting to cell complexes rules out only pathological spaces, not anything geometrically interesting. [remark: Why Cell Complexes] The restriction to cell complexes is not a loss of generality for the purposes of this course. Every space we want to study — spheres, projective spaces, surfaces, Lie groups — admits a cell complex structure. The structure is what makes the algebraic invariants computable: the chain groups in simplicial and cellular homology are free abelian groups generated by the cells, and the boundary maps have explicit combinatorial descriptions. [/remark] The remainder of the course develops two families of algebraic invariants — the *fundamental group* and *homology groups* — and applies them to prove the results sketched above. # 2. Definitions This chapter establishes the foundational vocabulary of the course and introduces the class of spaces — cell complexes — that we will study throughout. The preceding chapter explained why algebraic topology is useful: it translates hard topological non-existence questions into tractable algebraic ones. Before we can build that machinery, we need to pin down what kind of spaces and maps we are talking about, and equip ourselves with two technical lemmas that will be invoked repeatedly when constructing continuous maps by hand. ## Conventions on Maps and Continuity In point-set topology one can study arbitrary functions between spaces, whether or not they respect the topological structure. Algebraic topology has no use for discontinuous functions: every invariant we build — the fundamental group, covering spaces, homology — is designed to be preserved by continuous maps. For this reason the course adopts a universal convention from the outset. [definition: Map] In this course, **map** always means *continuous* map between topological spaces. Discontinuous functions are not considered. [/definition] This is not merely linguistic tidiness. Every time we say "there is a map $f: X \to Y$" or "construct a map", continuity is implicitly asserted. The convention frees us from writing "continuous" dozens of times per lecture. ### The Gluing Lemma A recurring construction in algebraic topology is to define a map piecewise on two overlapping or complementary subsets of a space, and then argue that the pieces assemble into a single continuous map. The gluing lemma is the tool that justifies this. [quotetheorem:1871] [citeproof:1871] [remark: Open Version] The gluing lemma holds equally with "closed" replaced throughout by "open". The proof is word-for-word the same, with the characterisation of continuity via preimages of open sets used in place of the closed-set version. [/remark] It is worth understanding why the two hypotheses — that the pieces be closed and that the cover be finite — are both necessary, not merely convenient. Suppose instead we try to glue along pieces that are not closed: split $\mathbb{R}$ as the closed piece $(-\infty, 0]$ and the open piece $(0, \infty)$, and define $f$ to be $-1$ on the first and $+1$ on the second. Each restriction is continuous on its piece, yet $f$ is discontinuous at $0$. Closedness of both pieces is what forces the boundary data to match. Finiteness is equally essential: every singleton $\{x\} \subset \mathbb{R}$ is closed, and the family $\{\{x\}\}_{x \in \mathbb{R}}$ is an infinite closed cover, but continuity on each singleton imposes no constraint at all — so if the lemma extended to infinite covers, every function would be continuous. The finite-union step in the proof is precisely where this fails. ### The Lebesgue Number Lemma The second technical tool controls how an open cover behaves on a compact metric space. Intuitively, on a compact space an open cover cannot become arbitrarily fine: there is always a positive scale $\delta$ below which every ball fits inside some cover element. [quotetheorem:952] [citeproof:952] The Lebesgue number lemma will be used in the construction of homotopies on compact spaces and in the proof of homotopy lifting for covering spaces, where we need to subdivide a path or homotopy into pieces each of which lies within a single open set of a cover. Two clarifications about the lemma's scope are important. First, compactness is genuinely necessary: the open cover $\{(1/(n+1), 1/n)\}_{n \geq 1}$ of $(0,1)$ has no Lebesgue number, since points near $0$ are not contained in any single cover element at a fixed positive scale. Second, the Lebesgue number $\delta$ depends on the specific cover $\mathcal{U}$, not only on the space $X$: a finer cover of the same compact space generally requires a smaller $\delta$. The lemma guarantees existence of some positive $\delta$ for each fixed cover, not a universal constant that works for all covers at once. ## Cell Complexes General topological spaces can be extraordinarily pathological. Even among compact Hausdorff spaces one encounters objects with bizarre local behaviour that resists any systematic analysis. In algebraic topology we restrict attention to spaces that are built by a clean, inductive procedure: starting from a discrete set of points and gluing in disks of increasing dimension. The resulting spaces are called **cell complexes**, and they are well-behaved enough that the invariants we develop will be computable. [motivation] The power of algebraic topology — the fundamental group, homology groups — comes from functoriality: a continuous map of spaces induces a homomorphism of groups. But for completely general spaces, the groups are often impossible to compute and theorems about them have so many exceptions that they become useless. The insight is to build spaces from the simplest possible pieces: points ($0$-cells), line segments ($1$-cells), disks ($2$-cells), and their higher-dimensional analogues. Each piece is a closed $n$-disk $D^n$, with boundary $\partial D^n = S^{n-1}$. By specifying how the boundary of each new disk attaches to the space already constructed, we get full control over the topology. The spaces we obtain this way are flexible enough to model surfaces, graphs, spheres, tori, projective spaces, and most spaces arising in geometry — yet regular enough that we can calculate their invariants explicitly. [/motivation] ### Cell Attachment The fundamental operation is attaching a single disk to an existing space by gluing its boundary sphere to a specified subset. Given a topological space $X$ and a map $f: S^{n-1} \to X$, we form a new space by taking the disjoint union of $X$ and a closed $n$-disk $D^n$ and then identifying each boundary point $x \in \partial D^n$ with its image $f(x) \in X$. The map $f$ is called the **attaching map**, and it completely determines how the new disk is sewn onto $X$; different choices of $f$ produce topologically distinct spaces from the same ingredients. [definition: Cell Attachment] Let $X$ be a topological space and let $f: S^{n-1} \to X$ be a map. The space obtained by **attaching an $n$-cell to $X$ along $f$** is the quotient \begin{align*} X \cup_f D^n = (X \amalg D^n) / {\sim}, \end{align*} where $\amalg$ denotes disjoint union and $\sim$ is the equivalence relation generated by $x \sim f(x)$ for all $x \in S^{n-1} \subseteq \partial D^n$. [/definition] The definition is concise but packs in a great deal. The key object is the quotient space: we are not choosing a subset of $X$ to glue onto, but specifying a continuous map from the boundary sphere into $X$, which may wrap, fold, or collapse the sphere in arbitrary ways. The attaching map $f$ is what gives cell complexes their flexibility — the same $n$-disk can be attached in wildly different ways to produce spaces with very different topological properties. [explanation: Geometric Meaning of Cell Attachment] The attaching map $f: S^{n-1} \to X$ selects a copy of the $(n-1)$-sphere sitting inside $X$ — this is the "target" where the boundary of the new disk will be glued. The quotient construction identifies each boundary point $x \in S^{n-1} = \partial D^n$ with its image $f(x) \in X$, while leaving the interior of $D^n$ untouched. Topologically, we are "plugging a disk" into $X$, with the boundary circle (or sphere) of the disk fused onto $X$ wherever $f$ sends it. For $n = 1$: a $1$-cell is an interval $D^1 = [-1, 1]$ with boundary $S^0 = \{-1, +1\}$. Attaching a $1$-cell means specifying two points in $X$ (possibly the same point) and connecting them with an edge. For $n = 2$: a $2$-cell is a disk $D^2$ with boundary circle $S^1$. Attaching a $2$-cell means specifying a loop (or more generally, a map of $S^1$) in $X$ and gluing the boundary of the disk onto that loop — in effect filling in the loop to make it contractible. [/explanation] [illustration:cell-attachment-2d] ### Cell Complexes Cell attachment is a single-step operation, but the power of the theory comes from iterating it. Starting from a finite discrete set of points — the $0$-skeleton — one attaches $1$-cells to connect them, then $2$-cells to fill in loops, then $3$-cells to fill in $2$-spheres, and so on. At each stage the space acquires new topology controlled by the attaching maps chosen in that step. The resulting inductive structure is what makes cell complexes both richly varied and computationally tractable: every topological feature of the space can be traced back to a specific cell and its attaching map. [definition: Cell Complex] A finite **cell complex** (or **CW complex**) is a space $X$ constructed by the following inductive procedure: 1. Begin with a finite discrete set $X^{(0)}$, called the **$0$-skeleton**. Its elements are the **$0$-cells** (vertices). 2. Given the **$(n-1)$-skeleton** $X^{(n-1)}$, form the **$n$-skeleton** $X^{(n)}$ by choosing a finite collection of attaching maps $\{f_\alpha: S^{n-1} \to X^{(n-1)}\}$ and setting \begin{align*} X^{(n)} = \left(X^{(n-1)} \amalg \coprod_\alpha D^n_\alpha\right) \Big/ \{x \sim f_\alpha(x)\}. \end{align*} Each $D^n_\alpha$ contributes one **$n$-cell** $e^n_\alpha$ (the image of the open disk $\operatorname{int}(D^n_\alpha)$ in the quotient). 3. Stop at $X = X^{(k)}$ for some $k$. The minimum such $k$ for which the process terminates is the **dimension** of $X$. To allow **infinite cell complexes**, remove the finiteness conditions and replace the stopping condition with: $X = \bigcup_{n \geq 0} X^{(n)}$, with the topology in which a set is closed if and only if its intersection with each $X^{(n)}$ is closed. [/definition] The definition encodes a remarkable amount of information in a compact form. Each skeleton $X^{(n)}$ is a quotient of a simpler space, and the transition from $X^{(n-1)}$ to $X^{(n)}$ is entirely controlled by the finite collection of attaching maps $\{f_\alpha\}$. This means that to understand a cell complex, it suffices to understand each attaching map — a great simplification compared to general topological spaces, where the global topology admits no such finite description. The remark below makes the partition structure explicit, which will be important when we define cellular homology. [remark: Open Cells Partition the Space] The open $n$-cells $\{e^n_\alpha\}$ across all dimensions form a partition of $X$: they are pairwise disjoint, and every point of $X$ belongs to exactly one open cell. This is the "CW" structure — Closure-finite and Weak topology — though in the finite case these conditions are automatic. [/remark] ### Examples of Cell Complexes [example: The $n$-Sphere] The sphere $S^n$ admits a minimal cell structure: one $0$-cell $e^0$ (a single point $p$) and one $n$-cell $e^n$. The attaching map $f: S^{n-1} \to \{p\}$ is the unique constant map sending every boundary point to $p$. Then \begin{align*} S^n = \{p\} \cup_f D^n. \end{align*} To see why this is correct: the quotient collapses the entire boundary $S^{n-1}$ of $D^n$ to the single point $p$, yielding $D^n / S^{n-1}$. This is indeed homeomorphic to $S^n$, because collapsing the boundary of a disk to a point produces a sphere — the interior of the disk becomes the open upper hemisphere, and the collapsed boundary becomes the single south pole. For $n = 1$: $S^1$ has one $0$-cell and one $1$-cell. The attaching map sends both endpoints $\{-1, +1\}$ to the single vertex, so the interval has both ends identified to the same point, giving a circle. For $n = 2$: $S^2$ has one $0$-cell and one $2$-cell. The attaching map collapses $S^1 = \partial D^2$ to a single point, producing a $2$-sphere. [/example] The sphere shows that even a very simple cell structure — a single cell in each relevant dimension — can produce a fundamental and geometrically rich space. More interesting spaces, however, need richer attaching maps that genuinely use the topology of the space already constructed. The torus is the first example where the attaching map for the top cell is non-trivial: it wraps the boundary circle around a figure-eight in a way that encodes the two independent cycles of the torus. The number of cells and the complexity of the attaching maps together determine the topological richness of the resulting space. [example: The Torus] The torus $T^2 = S^1 \times S^1$ has a cell structure with one $0$-cell, two $1$-cells, and one $2$-cell. Begin with a single vertex $v$. Attach two $1$-cells $a$ and $b$: the attaching map for each sends both endpoints of $[-1,1]$ to $v$, producing a wedge of two circles $S^1 \vee S^1$. This is the $1$-skeleton. Now attach a single $2$-cell with attaching map tracing the loop $aba^{-1}b^{-1}$ in $S^1 \vee S^1$ — that is, the map goes around $a$, then $b$, then $a$ backwards, then $b$ backwards. The resulting space is indeed homeomorphic to $T^2$. Geometrically, this corresponds to taking a square, labelling opposite edges as $a$ and $b$ (with matching orientations), and gluing them. [/example] [example: The Step-by-Step Construction of $X^{(2)}$] To make the inductive definition concrete, here is an explicit small example. Take $X^{(0)} = \{v_1, v_2, v_3, v_4\}$, a set of four points. For $X^{(1)}$: attach three $1$-cells connecting $v_1$-$v_2$, $v_2$-$v_3$, and $v_3$-$v_1$ (so attaching maps send the two endpoints of each interval to the respective pair of vertices), plus one loop at $v_4$ (attaching map sends both endpoints of the interval to $v_4$). The result is a triangle together with a loop. For $X^{(2)}$: attach one $2$-cell with attaching map going around the triangle $v_1 v_2 v_3$. This fills in the interior of the triangle, producing a disk (triangular face) together with the remaining loop at $v_4$. The final space $X^{(2)}$ has $4$ zero-cells, $4$ one-cells, and $1$ two-cell. [/example] ### A Non-Example: The Hawaiian Earring Not every space that looks like it might be built from cells actually is. The following example illustrates a fundamental failure and will later serve as a counterexample to several theorems. [example: Hawaiian Earring] The **Hawaiian Earring** $H$ is the subspace of $\mathbb{R}^2$ defined by \begin{align*} H = \bigcup_{n=1}^{\infty} C_n, \quad \text{where } C_n = \left\{(x,y) : x^2 + \left(y - \tfrac{1}{2^n}\right)^2 = \tfrac{1}{4^n}\right\} \end{align*} is the circle of radius $1/2^n$ centred at $(0, 1/2^n)$. All these circles pass through the origin $(0,0)$. At first glance, $H$ seems to be an infinite cell complex: each circle $C_n$ is a $1$-cell with both endpoints identified to the origin. However, $H$ is **not** a cell complex. The problem lies in the topology at the origin. In a genuine (infinite) cell complex built by attaching $1$-cells to the origin, the topology of the resulting space is the **weak topology**: a set $U$ is open if and only if $U$ intersects each cell in an open set within that cell. Under this topology, each circle can be given a neighbourhood that does not intersect any other circle, because the cells are topologically independent. In $H$, by contrast, the topology is inherited from $\mathbb{R}^2$, and the circles are not independent near the origin. To see this concretely: the top point of $C_n$ is $p_n = (0, 2/2^n) = (0, 1/2^{n-1})$. The sequence $(p_n)_{n \geq 1}$ satisfies $p_n \to (0, 0)$ in $\mathbb{R}^2$, since $|p_n| = 1/2^{n-1} \to 0$. So picking the "highest point" of each successive circle gives a sequence converging to the origin. If $H$ were a cell complex, the weak topology would mean we could assign a neighbourhood to each circle independently, and there would be no reason for a sequence that jumps between different cells to converge. More precisely: in an infinite cell complex with one $0$-cell and countably many $1$-cells, we could instead attach the $n$th circle with radius $n$ rather than $1/2^n$, making the top points diverge to infinity. But $H$ cannot be rearranged this way — its convergence properties are intrinsic to how the circles pile up at the origin, not merely an artifact of how we drew it. The Hawaiian Earring will reappear as a counterexample when we study the fundamental group and covering spaces. [/example] [illustration:hawaiian-earring] # 3. Homotopy and the Fundamental Group This chapter introduces the first and most fundamental algebraic invariant of a topological space: the fundamental group. The central strategy of algebraic topology is to translate topological problems — which are often hard — into algebraic problems, which are often tractable. The fundamental group is the first instance of this strategy: it assigns to each based topological space a group, in a way that is preserved by continuous maps and, crucially, by homotopy equivalences. By the end of this chapter, we will have defined $\pi_1(X, x_0)$, established that it is a genuine group, and shown that it is a homotopy invariant. ## Why Algebraic Invariants? [motivation] Recall the motivating question of algebraic topology: given two spaces $X$ and $Y$, how do we prove they are not homeomorphic? Producing a homeomorphism requires only that we exhibit a homeomorphism when spaces are equivalent, but disproving the existence of one requires invariants — properties that every homeomorphism must preserve. A clean first argument handles the case $\mathbb{R} \not\cong \mathbb{R}^2$. Remove a point from each space. After removing a point, $\mathbb{R}$ becomes disconnected (in fact, it splits into two open rays), while $\mathbb{R}^2 \setminus \{0\}$ remains path-connected. Since path-connectedness is a topological invariant, the two spaces cannot be homeomorphic. Unfortunately, this argument runs out of steam quickly. The spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ both remain path-connected after removing a point, so we cannot distinguish them this way. To go further, we need to detect something more subtle. When we remove the origin from $\mathbb{R}^2$, the punctured plane $\mathbb{R}^2 \setminus \{0\}$ is still path-connected, but something has genuinely changed: there are loops in $\mathbb{R}^2 \setminus \{0\}$ that cannot be continuously contracted to a point while staying in $\mathbb{R}^2 \setminus \{0\}$. Consider a circle centred at the origin: within the full plane $\mathbb{R}^2$ it can be shrunk to a point by scaling its radius to zero. But in $\mathbb{R}^2 \setminus \{0\}$, the origin is missing, and the circle is trapped around the hole — it cannot be contracted to a point without crossing the origin. [illustration:shrinking-loops-punctured-plane] This suggests a powerful idea: use the behaviour of loops under deformation as an invariant. Two spaces that look locally identical can be distinguished by asking whether loops in them can be deformed to points. The machinery that captures this is called homotopy, and the invariant it produces is the fundamental group $\pi_1(X, x_0)$. [/motivation] ## Homotopy The informal notion of "continuously deforming one map into another" is made precise by the definition of a homotopy. Throughout this section we write $I = [0, 1] \subset \mathbb{R}$ for the closed unit interval; it will appear constantly in what follows. [definition: Homotopy] Let $f, g: X \to Y$ be maps (continuous maps, as always in this course). A **homotopy** from $f$ to $g$ is a map \begin{align*} H: X \times I \to Y \end{align*} such that $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \in X$. We think of the parameter $t \in I$ as time: for each fixed $t$, the map $H(\,\cdot\,, t): X \to Y$ is an intermediate stage in the deformation from $f$ to $g$. If such an $H$ exists, we say $f$ is **homotopic** to $g$ and write $f \simeq g$. When we want to name the homotopy explicitly, we write $f \simeq_H g$. [/definition] The continuity requirement on $H$ is essential: it means the deformation itself varies continuously, ruling out "jumps" at any stage. Without it, any two maps between the same pair of spaces would be vacuously "deformable" into each other. Sometimes we need a constrained form of homotopy where some points of the domain are not allowed to move during the deformation. This arises naturally when deforming paths: we want the endpoints of a path to remain fixed. [definition: Homotopy rel A] Let $A \subseteq X$ be a subspace. We say $f$ is **homotopic to $g$ relative to $A$**, written $f \simeq g \operatorname{rel} A$, if there exists a homotopy $H: X \times I \to Y$ from $f$ to $g$ such that $H(a, t) = f(a) = g(a)$ for all $a \in A$ and all $t \in I$. [/definition] The notation $f \simeq g$ suggests homotopy is an equivalence relation, and this is indeed true. [quotetheorem:1872] [citeproof:1872] Transitivity is the most important of the three properties: it says we can compose homotopies end-to-end, forming a "concatenated" deformation that goes first from $f$ to $g$ and then from $g$ to $h$, all in one unit of time. The speed-doubling ($2t$ and $2t-1$) is a standard trick to achieve this within the parameter interval $[0,1]$. The equivalence relation structure is what makes it legitimate to form homotopy classes $[f]$ and to do algebra with them. The speed-doubling trick used in transitivity is not merely a technical convenience: it is precisely the same reparametrisation device that will reappear when we prove associativity of path concatenation. Understanding why transitivity needs this trick prepares us for the analogous construction in the path group laws below. ### Homotopy Equivalence of Spaces Having an equivalence relation on maps, we can promote the idea to spaces: two spaces are homotopy equivalent if there are maps back and forth whose compositions are homotopic to the identity maps. This is the "homotopy version" of homeomorphism, replacing equality of compositions with homotopy of compositions. [definition: Homotopy Equivalence] A map $f: X \to Y$ is a **homotopy equivalence** if there exists a map $g: Y \to X$ such that $f \circ g \simeq \operatorname{id}_Y$ and $g \circ f \simeq \operatorname{id}_X$. We call $g$ a **homotopy inverse** for $f$. If a homotopy equivalence $f: X \to Y$ exists, we say $X$ and $Y$ are **homotopy equivalent** and write $X \simeq Y$. [/definition] Every homeomorphism is a homotopy equivalence (take $g = f^{-1}$), but homotopy equivalence is much more flexible: it allows deformations that drastically change the visual shape of a space. The following examples illustrate this. [example: The Punctured Plane and the Circle] Let $X = S^1$ and $Y = \mathbb{R}^2 \setminus \{0\}$. There is a natural inclusion $i: S^1 \hookrightarrow \mathbb{R}^2 \setminus \{0\}$, and there is a retraction map $r: \mathbb{R}^2 \setminus \{0\} \to S^1$ defined by \begin{align*} r(y) = \frac{y}{\|y\|}, \end{align*} which projects every point radially onto the unit circle. By definition, $r(i(x)) = x$ for $x \in S^1$, so $r \circ i = \operatorname{id}_{S^1}$. It remains to show that $i \circ r \simeq \operatorname{id}_Y$. The map $i \circ r$ first projects a point $y$ onto $S^1$ and then includes it back; it is just the normalisation $y \mapsto y/\|y\|$. Define a homotopy $H: Y \times I \to Y$ by \begin{align*} H(y, t) = \frac{y}{t + (1 - t)\|y\|}. \end{align*} For each $t \in [0,1]$, the denominator $t + (1-t)\|y\|$ is a convex combination of $1$ and $\|y\|$, hence strictly positive on $Y$ (where $y \ne 0$), so $H$ is well-defined and continuous. At $t = 0$ we get $H(y, 0) = y/\|y\| = i \circ r(y)$, and at $t = 1$ we get $H(y, 1) = y = \operatorname{id}_Y(y)$. So $i \circ r \simeq \operatorname{id}_Y$, and $S^1 \simeq \mathbb{R}^2 \setminus \{0\}$. [/example] This example is striking: the two-dimensional punctured plane and the one-dimensional circle are homotopy equivalent. Homotopy equivalence simply does not care about dimension. What it does care about — and what both spaces share — is the presence of a "hole" that traps loops. [example: Contractibility of Euclidean Space] Let $Y = \mathbb{R}^n$ and $X = \{0\} =: *$. The inclusion $i: * \hookrightarrow \mathbb{R}^n$ and the unique map $r: \mathbb{R}^n \to *$ satisfy $r \circ i = \operatorname{id}_*$. The straight-line homotopy \begin{align*} H(y, t) = ty \end{align*} provides $i \circ r \simeq \operatorname{id}_{\mathbb{R}^n}$: at $t = 0$ every point maps to $0$, which is $i \circ r(y)$, and at $t = 1$ every point maps to itself. So $\mathbb{R}^n \simeq *$. [/example] A space homotopy equivalent to a single point is called **contractible**. From the perspective of homotopy theory, $\mathbb{R}^n$ is no more complicated than a single point. This may seem like we are throwing away too much structure, but the point is precisely to simplify: properties preserved by homotopy equivalence are easier to study, and complicated spaces can often be replaced by simpler homotopy-equivalent ones. [definition: Contractible Space] A space $X$ is **contractible** if $X \simeq *$, i.e., if the identity map $\operatorname{id}_X$ is homotopic to a constant map $X \to \{x_0\}$ for some $x_0 \in X$. [/definition] Homotopy equivalence of spaces is itself an equivalence relation. This follows from a key lemma about composition of homotopic maps. [quotetheorem:1873] [citeproof:1873] This lemma is what allows us to define functors on the homotopy category. It says that homotopy is a **congruence** for composition: $f_0 \simeq f_1$ implies $h \circ f_0 \simeq h \circ f_1$ and $f_0 \circ k \simeq f_1 \circ k$ for any composable maps $h$ and $k$. Without this, passing from maps to homotopy classes would be incoherent — the composition of two homotopy classes $[g] \circ [f]$ would not be well-defined, since a different representative $f_1 \simeq f_0$ might yield a different composite. The lemma guarantees that this cannot happen, making the homotopy category well-defined. [quotetheorem:1874] [citeproof:1874] The equivalence relation on spaces is what justifies speaking of **homotopy types** as genuine equivalence classes rather than mere pairwise relationships. Transitivity in particular requires the composition lemma established above: without knowing that homotopic maps compose to homotopic maps, the transitivity argument would break down at the step where $f \circ g \simeq \operatorname{id}_Y$ is used inside a larger composition. Spaces sharing a homotopy type share all homotopy invariants — including $\pi_0$, $\pi_1$, and all higher homotopy groups and homology groups that will appear later in the course. ### Retractions and Deformation Retractions Two related notions refine homotopy equivalence by tracking how a subspace sits inside a larger space. They will appear throughout the course. [definition: Retraction] Let $A \subseteq X$ be a subspace. A **retraction** is a map $r: X \to A$ such that $r \circ i = \operatorname{id}_A$, where $i: A \hookrightarrow X$ is the inclusion. If such an $r$ exists, $A$ is called a **retract** of $X$. [/definition] A retraction sends every point of $X$ to $A$ while fixing $A$ pointwise. The condition $r \circ i = \operatorname{id}_A$ means that no point of $A$ is moved. The retraction says, loosely, that $A$ is no more complex than $X$. [definition: Deformation Retraction] A retraction $r: X \to A$ is a **deformation retraction** if moreover $i \circ r \simeq \operatorname{id}_X$. It is a **strong deformation retraction** if $i \circ r \simeq \operatorname{id}_X \operatorname{rel} A$, i.e., the homotopy keeps points of $A$ fixed throughout. [/definition] When $A$ is a deformation retract of $X$, the spaces $A$ and $X$ are homotopy equivalent: the inclusion $i: A \hookrightarrow X$ and the retraction $r: X \to A$ are mutual homotopy inverses. This is the typical way homotopy equivalences arise in practice. [example: Contractible Spaces and Point Retracts] Let $X$ be any space and $A = \{x_0\} \subseteq X$ a single point. The constant map $r: X \to \{x_0\}$ is always a retraction. If $X$ is contractible, then $\{x_0\}$ is a deformation retract of $X$: the contracting homotopy witnesses $i \circ r \simeq \operatorname{id}_X$. [/example] ## Paths With the notion of homotopy established, we turn to the central objects through which algebraic topology detects the structure of a space: paths and loops. [definition: Path] A **path** in a space $X$ is a continuous map $\gamma: I \to X$. If $\gamma(0) = x_0$ and $\gamma(1) = x_1$, we say $\gamma$ is a path from $x_0$ to $x_1$ and write $\gamma: x_0 \rightsquigarrow x_1$. When $\gamma(0) = \gamma(1)$, the path is called a **loop** based at $x_0$. [/definition] A path need not be injective: it can backtrack, self-intersect, or remain stationary. Only continuity is required. The basic idea is to use paths to detect the shape of a space by asking which loops can be contracted to a point. To do algebra with paths, we need to be able to compose them. The following three operations are the building blocks. [definition: Concatenation of Paths] If $\gamma_1: x_0 \rightsquigarrow x_1$ and $\gamma_2: x_1 \rightsquigarrow x_2$ are paths in $X$ with $\gamma_1(1) = \gamma_2(0)$, their **concatenation** is the path $\gamma_1 \cdot \gamma_2: x_0 \rightsquigarrow x_2$ defined by \begin{align*} (\gamma_1 \cdot \gamma_2)(t) = \begin{cases} \gamma_1(2t) & 0 \leq t \leq \tfrac{1}{2}, \\ \gamma_2(2t - 1) & \tfrac{1}{2} \leq t \leq 1. \end{cases} \end{align*} This is continuous by the Gluing Lemma. [/definition] Note that concatenation is left-to-right: $\gamma_1 \cdot \gamma_2$ means first traverse $\gamma_1$, then $\gamma_2$. This is the opposite convention from function composition. Keeping this in mind prevents errors when working with fundamental groups. [definition: Inverse of a Path] The **inverse** of a path $\gamma: I \to X$ is the path $\gamma^{-1}: I \to X$ defined by \begin{align*} \gamma^{-1}(t) = \gamma(1 - t). \end{align*} Geometrically, $\gamma^{-1}$ traverses the same image as $\gamma$ but in the opposite direction. [/definition] [definition: Constant Path] The **constant path** at a point $x \in X$ is the path $c_x: I \to X$ defined by $c_x(t) = x$ for all $t \in I$. [/definition] These three operations — concatenation, inverse, and constant path — resemble the data of a group. However, they do not form a group over paths themselves: the concatenation $\gamma \cdot \gamma^{-1}$ is not literally the constant path, only a path that goes somewhere and comes back. Associativity likewise fails at the level of individual paths (since reparametrisation changes the path as a function, even if the image is the same). To extract genuine algebraic structure, we must pass to equivalence classes under homotopy. ### Path Components Before doing that, we can already associate a useful invariant to a space using paths alone. [definition: Path Components] Define a relation on $X$ by $x \sim y$ if there exists a path from $x$ to $y$. The concatenation, inverse, and constant path show that $\sim$ is an equivalence relation. The equivalence classes are called **path components** of $X$. The set of path components is denoted $\pi_0(X)$. [/definition] The set $\pi_0(X)$ is our first example of a functor: it assigns a set to each space and a function between sets to each continuous map. [quotetheorem:1875] [citeproof:1875] It is worth noting that $\pi_0(X)$ is only a **set**, not a group. It captures path-connectedness — how many connected pieces the space has — but carries no information about loops. In particular, $\pi_0$ cannot distinguish the circle $S^1$ from the interval $[0,1]$: both are path-connected, so both have $|\pi_0| = 1$. This inadequacy is precisely what motivates the next functor, $\pi_1$, which carries a group structure and detects whether loops in a path-connected space can be contracted. [remark: Homotopy Equivalences Induce Bijections on pi-zero] If $f: X \to Y$ is a homotopy equivalence with homotopy inverse $g$, then properties (1)–(3) imply that $\pi_0(f)$ and $\pi_0(g)$ are inverse bijections, so $\pi_0(f): \pi_0(X) \to \pi_0(Y)$ is a bijection. In particular, homotopy equivalent spaces have the same number of path components. [/remark] This gives us a clean proof of a simple but illustrative result. [example: Non-contractibility of Two Points] The two-point space $X = \{-1, 1\}$ (with the discrete topology) is not contractible. Indeed, $|\pi_0(X)| = 2$ while $|\pi_0(*)| = 1$. Since any homotopy equivalence $X \simeq *$ would induce a bijection $\pi_0(X) \to \pi_0(*)$, no such equivalence exists. [/example] Most spaces we care about are path-connected, meaning $\pi_0(X)$ has exactly one element, and in this case $\pi_0$ gives no further information. We need a finer invariant, one that detects loops rather than just path components. ### Homotopy of Paths The key step is to pass from individual paths to homotopy classes of paths. For this we use homotopy relative to the endpoints. [definition: Homotopy of Paths] Two paths $\gamma, \gamma': I \to X$ with the same endpoints ($\gamma(0) = \gamma'(0)$ and $\gamma(1) = \gamma'(1)$) are **homotopic as paths** if they are homotopic rel $\{0, 1\}$. That is, there exists a map $H: I \times I \to X$ with $H(s, 0) = \gamma(s)$, $H(s, 1) = \gamma'(s)$, and $H(0, t) = \gamma(0)$, $H(1, t) = \gamma(1)$ for all $s, t \in I$. We again write $\gamma \simeq \gamma'$. [/definition] Fixing the endpoints is essential: if we allowed them to move, any path could be contracted to a constant path (just pull both endpoints together), making the notion useless. Path homotopy is compatible with concatenation: if $\gamma_1 \simeq \gamma_1'$ and $\gamma_2 \simeq \gamma_2'$ (with compatible endpoints), then $\gamma_1 \cdot \gamma_2 \simeq \gamma_1' \cdot \gamma_2'$. This is proved by constructing a homotopy that runs $H_1$ on the first half of the parameter square and $H_2$ on the second half. The following proposition is the algebraic backbone for the definition of the fundamental group. It says that paths up to homotopy satisfy all the axioms needed to form a group, provided we restrict to a fixed basepoint. [quotetheorem:1877] [citeproof:1877] A crucial observation: all three laws — associativity, identity, and inverse — hold only **up to homotopy**, not as equalities of paths. At the level of actual functions $I \to X$, even associativity fails: $(\gamma_0 \cdot \gamma_1) \cdot \gamma_2$ and $\gamma_0 \cdot (\gamma_1 \cdot \gamma_2)$ parametrise the same image at different speeds, making them distinct maps. It is precisely for this reason that the fundamental group must be defined on **homotopy classes** of loops rather than on loops themselves. The passage to equivalence classes is not a convenience but a necessity. ## The Fundamental Group We now have all the ingredients to build the fundamental group. The group law above fails to give a genuine group on all paths because we can only concatenate paths when endpoints match. The solution is to fix a **basepoint** $x_0 \in X$ and restrict attention to loops at $x_0$ — paths that begin and end at $x_0$. Any two such loops can always be concatenated. [definition: Fundamental Group] Let $X$ be a space and $x_0 \in X$. The **fundamental group** of $X$ based at $x_0$, denoted $\pi_1(X, x_0)$, is the set of homotopy classes of loops based at $x_0$: \begin{align*} \pi_1(X, x_0) = \{ [\gamma] : \gamma: I \to X \text{ continuous},\ \gamma(0) = \gamma(1) = x_0 \} / {\simeq}. \end{align*} The group structure is: - **Multiplication**: $[\gamma_0] \cdot [\gamma_1] = [\gamma_0 \cdot \gamma_1]$ (concatenation of loops). - **Identity**: $e = [c_{x_0}]$ (the constant loop at $x_0$). - **Inverse**: $[\gamma]^{-1} = [\gamma^{-1}]$ (the reversed loop). [/definition] [quotetheorem:1878] [citeproof:1878] This theorem depends crucially on restricting to **loops** — paths satisfying $\gamma(0) = \gamma(1) = x_0$. Without a fixed basepoint, concatenation of two arbitrary paths $\gamma_0: x_0 \rightsquigarrow x_1$ and $\gamma_1: x_2 \rightsquigarrow x_3$ is undefined whenever $x_1 \ne x_2$. Fixing the basepoint ensures that every loop can be concatenated with every other loop, making the multiplication well-defined for all pairs of elements. This means $\pi_1$ is intrinsically **basepoint-dependent**, and the next subsection studies precisely what happens when the basepoint is changed. The fundamental group is genuinely difficult to compute from scratch for most spaces. The reason is that to show a loop is not contractible, one needs to prove the absence of a homotopy, which requires knowing something about the global topology of the space. We will develop machinery — covering spaces, the Seifert–van Kampen theorem — in later chapters to do these computations. For now, we establish the formal properties of $\pi_1$. ### Functoriality Just as $\pi_0$ turned maps into functions between sets, $\pi_1$ turns based maps into group homomorphisms. This is the key functoriality property. [definition: Based Space] A **based space** is a pair $(X, x_0)$ consisting of a topological space $X$ and a distinguished point $x_0 \in X$, the **basepoint**. A **map of based spaces** $f: (X, x_0) \to (Y, y_0)$ is a continuous map $f: X \to Y$ with $f(x_0) = y_0$. A **based homotopy** is a homotopy rel $\{x_0\}$. [/definition] [quotetheorem:1879] [citeproof:1879] The hypothesis that $f(x_0) = y_0$ is essential: without it, $f \circ \gamma$ would be a loop based at $f(x_0)$, not at a fixed basepoint in $Y$, and the induced map $\pi_1(X, x_0) \to \pi_1(Y, y_0)$ would not even be defined as stated. For maps that do not preserve basepoints, one can still say something, but only after choosing a path from $f(x_0)$ to $y_0$ — which introduces an ambiguity depending on the path chosen. This is precisely the content of the change-of-basepoint theorem in the next subsection, which explains exactly how the induced map changes when the basepoint is moved. In categorical language, $\pi_1$ is a functor from the category of based topological spaces to the category of groups. This is the formal content of the phrase "fundamental group is a topological invariant." ### Changing the Basepoint Defining $\pi_1(X, x_0)$ requires the choice of a basepoint $x_0$, but a space should have intrinsic algebraic properties independent of this choice. We now examine what happens when the basepoint changes. The fundamental group $\pi_1(X, x_0)$ only detects the path component containing $x_0$: every loop based at $x_0$ stays within that component. For this reason, we typically assume $X$ is path-connected when working with fundamental groups. Assuming path-connectedness, different basepoints yield isomorphic fundamental groups. The isomorphism is constructed by conjugating loops with a path connecting the two basepoints. [quotetheorem:1880] [citeproof:1880] The main consequence is that for a path-connected space $X$, all the fundamental groups $\pi_1(X, x_0)$ are isomorphic to one another, so we often write simply $\pi_1(X)$. However, the isomorphism $u_\#$ depends on the choice of path $u$. If we change $u$, we obtain a different isomorphism — specifically, one that differs by conjugation (by property 5 applied when $x_1 = x_0$). There is therefore no canonical identification between fundamental groups at different basepoints, and it is not meaningful to say that a specific element $\alpha \in \pi_1(X, x_0)$ "corresponds to" a specific element in $\pi_1(X, x_1)$. ### Homotopy Invariance The key theorem is that the fundamental group is invariant under homotopy equivalence. This is not obvious: a homotopy equivalence $f: X \to Y$ need not preserve the basepoint, so the induced map on fundamental groups involves different basepoints. The following lemma bridges the gap between based and unbased homotopies, and is a model proof that we will use again. [quotetheorem:1881] [citeproof:1881] This proof strategy — lifting a problem from a complicated space $X$ to the tractable square $I \times I$ and using convexity — is a recurring technique in homotopy theory. With this lemma in hand, the main theorem follows. [quotetheorem:1882] [citeproof:1882] This theorem is the payoff for all the foundational work done above. It guarantees that the fundamental group is a genuine topological invariant: if $X$ and $Y$ are homotopy equivalent, their fundamental groups are isomorphic. In particular, if two spaces have non-isomorphic fundamental groups, they cannot be homotopy equivalent (and so in particular cannot be homeomorphic). ### Simply Connected Spaces A space with $\pi_1(X, x_0) = \{e\}$ is the simplest possible case, and it deserves a name. [definition: Simply Connected Space] A space $X$ is **simply connected** if it is path-connected and $\pi_1(X, x_0) \cong \{e\}$ for some (equivalently, any) basepoint $x_0 \in X$. [/definition] [example: Contractible Spaces are Simply Connected] Any contractible space $X$ is simply connected. Indeed, $X \simeq *$, and since $\pi_1(*, *) = \{e\}$ (the only loop at a single point is the constant loop), the homotopy invariance theorem gives $\pi_1(X, x_0) \cong \{e\}$. In particular, $\mathbb{R}^n$ is simply connected for all $n \geq 0$. [/example] Simply connected spaces have a clean characterisation in terms of uniqueness of path homotopy classes. [quotetheorem:1883] [citeproof:1883] This characterisation is both natural and useful: simply connected spaces are exactly those in which paths between any two points are unique up to homotopy, leaving no room for loops that are not null-homotopic. [remark: Outlook] We have defined the fundamental group and established its formal properties, but we have not yet computed it for any interesting space. To show that $\pi_1(S^1) \cong \mathbb{Z}$ — the paradigmatic non-trivial example — we need the theory of covering spaces, which comes in the next chapter. Once that is in place, the connection between covering spaces and subgroups of $\pi_1$ will give us powerful computational tools: the Seifert–van Kampen theorem will allow us to compute fundamental groups of spaces built from simpler pieces. [/remark] # 4. Covering Spaces The previous chapter developed the fundamental group $\pi_1(X, x_0)$ as an algebraic invariant of a topological space. But what does $\pi_1$ actually act on? Groups are not abstract objects that exist in isolation — they are symmetries of something. The machinery we built detected loops, concatenations, and homotopy classes, but we have yet to see these groups doing anything geometric. This chapter answers the question by constructing covering spaces: the natural objects on which fundamental groups act. Along the way, we will compute our first example of a fundamental group with more than one element, prove Brouwer's fixed point theorem in dimension two, and establish a precise dictionary between the subgroup lattice of $\pi_1(X, x_0)$ and the category of covering spaces over $X$ — the Galois correspondence. ## Covering Spaces and Lifting Here is a concrete obstacle: suppose you want to prove that $\pi_1(S^1, 1)$ is not the trivial group. How would you do it? You cannot exhibit an explicit loop that is not null-homotopic by hand — any loop in $S^1$ can be dragged around, and it is not at all clear when two loops are homotopic. We need a mechanism that converts the question "is this loop null-homotopic?" into something more algebraically legible. Covering spaces provide exactly this mechanism, by allowing us to lift loops from a complicated space into a simpler one where homotopies are easier to track. [definition: Evenly Covered Neighbourhood] An open set $U \subseteq X$ is **evenly covered** by $p: \tilde{X} \to X$ if \begin{align*} p^{-1}(U) \cong \coprod_{\alpha \in \Lambda} V_\alpha, \end{align*} where each $V_\alpha \subseteq \tilde{X}$ is open, and the restriction $p|_{V_\alpha}: V_\alpha \to U$ is a homeomorphism for each $\alpha \in \Lambda$. [/definition] The sets $V_\alpha$ are called the **sheets** lying over $U$. Each one is an honest copy of $U$, sitting in $\tilde{X}$, and the map $p$ restricts to a homeomorphism on each sheet. The index set $\Lambda$ may be finite or infinite, but the point is that locally, $\tilde{X}$ looks like many identical replicas of $X$. [definition: Covering Space] A **covering space** of $X$ is a pair $(\tilde{X}, p)$ where $p: \tilde{X} \to X$ is a continuous map such that every point $x \in X$ has an open neighbourhood $U$ that is evenly covered by $p$. [/definition] Whether we require $p$ to be surjective is a matter of convention. For path-connected $X$ — the case we care about — surjectivity is automatic once any point has a preimage, so we will not worry about it. [illustration:covering-space-sheets] [example: The Real Line Covers the Circle] Define $p: \mathbb{R} \to S^1 \subseteq \mathbb{C}$ by $t \mapsto e^{2\pi i t}$. This is a covering map. To check the definition, take any point $z_0 = e^{2\pi i t_0} \in S^1$ and let $U$ be the arc $\{e^{2\pi i t} : t \in (t_0 - \varepsilon, t_0 + \varepsilon)\}$ for small $\varepsilon > 0$. The preimage is \begin{align*} p^{-1}(U) = \coprod_{n \in \mathbb{Z}} (t_0 - \varepsilon + n, \, t_0 + \varepsilon + n), \end{align*} a disjoint union of open intervals in $\mathbb{R}$, each of which maps homeomorphically onto $U$ by $p$. Thus $U$ is evenly covered, and $p^{-1}(1) = \mathbb{Z}$. [/example] [example: Degree-$n$ Covering of the Circle] For any nonzero integer $n$, the map $p_n: S^1 \to S^1$ defined by $z \mapsto z^n$ is an $n$-sheeted covering. Here the preimage of $1 \in S^1$ consists of the $n$ roots of unity: $p_n^{-1}(1) = \{e^{2\pi i k/n} : k = 0, 1, \ldots, n-1\}$. This is very different from the previous example where infinitely many sheets appear: the covering $p: \mathbb{R} \to S^1$ has a preimage of every point consisting of $\mathbb{Z}$-many points, whereas $p_n$ produces only $n$. [/example] [example: The Double Cover of the Real Projective Plane] The real projective plane $\mathbb{RP}^2$ is defined as the quotient $S^2/{\sim}$ where antipodal points satisfy $x \sim -x$. The quotient map $p: S^2 \to \mathbb{RP}^2$ is a covering map. Given any point $[x_0] \in \mathbb{RP}^2$, choose a small open hemisphere $U$ around $x_0$. Then $p^{-1}(U)$ consists of two disjoint open sets — the hemisphere around $x_0$ and the antipodal hemisphere around $-x_0$ — each mapping homeomorphically onto the image. This is thus a 2-sheeted covering. [/example] These examples illustrate a key intuition: $\tilde{X}$ is locally indistinguishable from $X$, but globally it may be "unwrapped" into something simpler. Before exploiting this, we need a mechanism to move maps from $X$ up to $\tilde{X}$. [definition: Lift] Let $f: Y \to X$ be a continuous map and $p: \tilde{X} \to X$ a covering space. A **lift** of $f$ is a continuous map $\tilde{f}: Y \to \tilde{X}$ such that $p \circ \tilde{f} = f$, i.e., the diagram \begin{align*} \tilde{X} & \\ \tilde{f} \nearrow \quad & \downarrow p \\ Y \xrightarrow{f} \quad & X \end{align*} commutes. [/definition] A lift is precisely a choice of "which sheet" to live in. Since each sheet is a homeomorphic copy of $X$, lifting means passing from $X$ to one particular replica in $\tilde{X}$. The first key fact about lifts is that they are essentially unique once a single point is fixed. [quotetheorem:1885] [citeproof:1885] The uniqueness theorem says that a lift is entirely determined by its value at a single point. Once we decide which sheet $\tilde{f}$ starts in, the rest of the map is forced. This is exactly the intuition we wanted: once you choose which copy of $X$ you are working in, continuity forces you to stay in that copy (locally), and since $Y$ is connected, this propagates globally. The uniqueness is now established. The existence question is subtler: given $f: Y \to X$ and a covering $p: \tilde{X} \to X$, when can we actually find a lift? A general criterion will come later, but the crucial special case — lifting paths and homotopies — always works. [quotetheorem:1886] [citeproof:1886] The path lifting lemma is a special case of the homotopy lifting lemma, which handles arbitrary homotopies, not just single paths. The homotopy lifting lemma is proved by promoting path lifting: for each fixed $y \in Y$, we lift the path $H(y, -)$; then we use the compactness of $I$ and uniqueness of lifts to paste these individual path lifts together into a jointly continuous map $\tilde{H}: Y \times I \to \tilde{X}$. The strategy is: (1) use compactness of $I$ to control the path lifting on small neighbourhoods in $Y$, (2) use uniqueness to guarantee that overlapping local lifts agree, and (3) deduce global continuity. [quotetheorem:1887] The proof of the full homotopy lifting lemma requires the same connectedness argument as path lifting, applied fibrewise, together with a compactness argument to ensure the local path lifts can be assembled. Uniqueness is again an immediate consequence of the uniqueness of lifts theorem. Having lifted homotopies, we can ask whether a homotopy of paths lifts to a homotopy of paths — that is, whether the lifted homotopy also preserves the basepoint at both ends. This is the crucial strengthening needed for working with $\pi_1$. [quotetheorem:1888] [citeproof:1888] This theorem has a consequence that immediately tells us something about the structure of covering spaces. In a path-connected space $X$, it would be strange if different base points had different-sized preimages — why should points in the same connected component be treated differently? The theorem rules this out. [quotetheorem:1889] [citeproof:1889] This justifies the following definition, which attaches a clean numerical invariant to each covering. [definition: $n$-Sheeted Covering] A covering space $p: \tilde{X} \to X$ of a path-connected space $X$ is called **$n$-sheeted** if $|p^{-1}(x)| = n$ for any (equivalently, all) $x \in X$. If $|p^{-1}(x)| = \infty$ for all $x$, we say the covering is infinite-sheeted. [/definition] The number of sheets is the topological analogue of the index of a subgroup. This parallel will become precise in the Galois correspondence. For now, let us extract the key algebraic consequence of the lifting theory. [quotetheorem:1890] [citeproof:1890] This injectivity means that the covering map $p$ identifies $\pi_1(\tilde{X}, \tilde{x}_0)$ with a subgroup of $\pi_1(X, x_0)$, namely the image $p_* \pi_1(\tilde{X}, \tilde{x}_0)$. We have therefore constructed a map from covering spaces to subgroups of $\pi_1$. The Galois correspondence, developed in the final section, will show this is a bijection. ### The Monodromy Action We originally asked: what does $\pi_1(X, x_0)$ act on? We now have the answer. Consider the fibre $p^{-1}(x_0)$. Given a based loop $\gamma: I \to X$ at $x_0$ and a point $\tilde{x}_0 \in p^{-1}(x_0)$, the path lifting lemma gives a unique lift $\tilde{\gamma}$ starting at $\tilde{x}_0$; its endpoint $\tilde{\gamma}(1)$ is another point of $p^{-1}(x_0)$. By the lifted path homotopy theorem, homotopic loops produce the same endpoint. We therefore have a well-defined action: \begin{align*} \tilde{x}_0 \cdot [\gamma] := \tilde{\gamma}(1) \in p^{-1}(x_0). \end{align*} Composing two loops shows this is a right action of $\pi_1(X, x_0)$ on $p^{-1}(x_0)$, called the **monodromy action**. The orbit and stabiliser of this action carry precise geometric meaning. [quotetheorem:1891] [citeproof:1891] The theorem has an important consequence: if $\tilde{X}$ is path connected and $p^{-1}(x_0)$ has more than one element, then $\pi_1(X, x_0)$ is nontrivial, since the orbit of any $\tilde{x}_0$ under a trivial group would consist of just $\tilde{x}_0$ itself. The real line $\mathbb{R}$ covering $S^1$ already tells us that $\pi_1(S^1, 1)$ is infinite — though we have not yet determined its group structure. [definition: Universal Cover] A covering map $p: \tilde{X} \to X$ is a **universal cover** if $\tilde{X}$ is simply connected, i.e., path connected and $\pi_1(\tilde{X}, \tilde{x}_0) = \{e\}$ for some (equivalently, all) $\tilde{x}_0 \in \tilde{X}$. [/definition] When the covering is universal, the stabiliser subgroup in (2) is trivial, so (3) gives a bijection $\pi_1(X, x_0) \to p^{-1}(x_0)$. The elements of $\pi_1(X, x_0)$ are in bijection with the sheets of the covering — and the group structure of $\pi_1$ is encoded in how these sheets permute under the monodromy action. This is the bridge we need to compute $\pi_1(S^1, 1)$. [quotetheorem:1893] The bijection $\ell$ depends on the choice of $\tilde{x}_0$: changing $\tilde{x}_0$ conjugates the identification. This is an early hint of the Galois correspondence, where basepoint changes correspond to conjugation of subgroups. ## The Fundamental Group of the Circle and Its Applications We now have the tools to compute $\pi_1(S^1, 1)$ and to turn that computation into hard theorems. The key covering to use is $p: \mathbb{R} \to S^1$, $t \mapsto e^{2\pi i t}$. The real line $\mathbb{R}$ is simply connected (it is contractible), so this is a universal cover, and the universal cover bijection gives $\pi_1(S^1, 1) \leftrightarrow p^{-1}(1) = \mathbb{Z}$. We have established that $\pi_1(S^1, 1)$ has the same cardinality as $\mathbb{Z}$. But cardinality alone does not determine the group. To complete the computation, we need to verify that the bijection $\ell: \pi_1(S^1, 1) \to \mathbb{Z}$ is actually a group homomorphism. [quotetheorem:1894] [citeproof:1894] The argument is more than just computation: it reveals the essential role of the covering space as a "linearisation." A loop in $S^1$ can wind arbitrarily, making direct homotopies difficult to write down. Lifting to $\mathbb{R}$ straightens everything out — any path in $\mathbb{R}$ from $0$ to $n$ is homotopic to the straight path $t \mapsto nt$, since $\mathbb{R}$ is simply connected. We then project this homotopy back to $S^1$ to conclude. Notice that the hypothesis "$\mathbb{R}$ is simply connected" is not merely a convenience — without it we could not produce the homotopy upstairs, and the argument would fail. Any space homotopy equivalent to $S^1$, such as $\mathbb{C} \setminus \{0\}$, also has fundamental group $\mathbb{Z}$, since the fundamental group is a homotopy invariant. The computation of $\pi_1(S^1)$ immediately yields a classical result. [definition: Winding Number] Let $\alpha: S^1 \to \mathbb{C} \setminus \{0\}$ be a closed curve. Since $\mathbb{C} \setminus \{0\} \simeq S^1$, the map $\alpha$ induces a group homomorphism $\alpha_*: \pi_1(S^1, 1) \to \pi_1(\mathbb{C} \setminus \{0\}, \alpha(1)) \cong \mathbb{Z}$. Every homomorphism $\mathbb{Z} \to \mathbb{Z}$ is of the form $k \mapsto nk$ for a unique $n \in \mathbb{Z}$. This integer $n$ is the **winding number** of $\alpha$ around the origin. [/definition] [example: Winding Numbers Concretely] The loop $\alpha: t \mapsto e^{2\pi i t}$ traverses $S^1$ once counterclockwise. Its winding number is $1$. The loop $\beta: t \mapsto e^{-4\pi i t}$ traverses $S^1$ twice clockwise, so its winding number is $-2$. A figure-eight curve that winds once counterclockwise around the origin and once clockwise has winding number $0$ — but it is not null-homotopic in $\mathbb{C} \setminus \{0\}$, since the two loops do not cancel in $\pi_1$. This example shows that zero winding number does not imply null-homotopy in general; for simple loops around the origin, however, the winding number completely classifies homotopy classes. [/example] The next application of $\pi_1(S^1, 1) \cong \mathbb{Z}$ is a proof of Brouwer's fixed point theorem in dimension two. The argument does not generalise directly to higher dimensions using $\pi_1$ alone, but the two-dimensional case already contains the key idea. [quotetheorem:1895] [citeproof:1895] The proof has a distinctive structure: we assume the geometric statement fails, construct a retraction of $D^2$ onto its boundary $S^1$, and then derive an algebraic contradiction — no group homomorphism can factor the identity through the trivial group. The hypothesis that $D^2$ is contractible (hence $\pi_1(D^2) = \{e\}$) is essential: it is precisely what makes the middle group trivial. If $D^2$ were replaced by an annulus, which has $\pi_1 \cong \mathbb{Z}$, the algebraic argument would not produce a contradiction, and indeed the analogous fixed-point theorem fails for annuli. The same approach in dimension three would require knowing $\pi_1(S^2)$. Since $S^2$ is simply connected — a fact we will be able to verify later using van Kampen's theorem or simplicial methods — the two-dimensional fundamental group gives no information. Higher-dimensional Brouwer theorems require homology, developed in later chapters. ## Universal Covers We introduced universal covers as covering spaces whose total space is simply connected. We have seen one example ($\mathbb{R} \to S^1$) and used it decisively. But do universal covers always exist? What do they look like in general? Before asking for existence, we should ask what having a universal cover implies about $X$. Suppose $p: \tilde{X} \to X$ exists with $\tilde{X}$ simply connected. Take any $x_0 \in X$ and any evenly covered neighbourhood $U$ of $x_0$. Any loop $\gamma$ in $X$ based at $x_0$ and contained in $U$ lifts to a loop $\tilde{\gamma}$ in $\tilde{X}$ (starting in the sheet $\tilde{U}$ over $U$). Since $\tilde{X}$ is simply connected, $\tilde{\gamma}$ is null-homotopic in $\tilde{X}$. Projecting the null-homotopy down gives a null-homotopy of $\gamma$ in $X$ — but the homotopy need not stay in $U$; it can wander anywhere in $X$. So the condition we need on $X$ is not that $U$ itself be simply connected, but only that every loop in $U$ based at $x_0$ is contractible in $X$. [definition: Semi-Locally Simply Connected] A topological space $X$ is **semi-locally simply connected** if for every $x \in X$ there exists an open neighbourhood $U$ of $x$ such that every loop in $U$ based at $x$ is null-homotopic in $X$ (though not necessarily within $U$). [/definition] [definition: Locally Path Connected] A topological space $X$ is **locally path connected** if for every $x \in X$ and every open neighbourhood $V$ of $x$, there exists an open path-connected set $U$ with $x \in U \subseteq V$. [/definition] Path connectedness alone does not imply local path connectedness. The standard counterexample is the topologist's sine curve: the closure of $\{(x, \sin(1/x)) : x > 0\}$ in $\mathbb{R}^2$ is path connected (in fact, connected) but not locally path connected at any point of the segment $\{0\} \times [-1, 1]$, since every small neighbourhood of such a point has infinitely many connected components. [quotetheorem:1896] [citeproof:1896] The constructive proof is worth dwelling on. It tells us something profound: the universal cover of $X$ is the space of homotopy classes of paths starting from $x_0$. Two paths represent the same point of $\tilde{X}$ when they are homotopic as paths. This means that the universal cover "remembers" all the ways you can travel from $x_0$ — it is the space of "paths modulo homotopy." [example: Universal Cover of the Torus] The torus $T = S^1 \times S^1$ has universal cover $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$, covered by $p \times p: \mathbb{R}^2 \to S^1 \times S^1$. This means $\pi_1(T) \cong \mathbb{Z} \times \mathbb{Z}$. To understand this geometrically, think of the torus as the unit square $[0,1]^2$ with opposite sides identified. The universal cover tiles the plane: $\mathbb{R}^2$ looks locally like the torus, but globally it is just flat space. Living in the torus feels like living in $\mathbb{R}^2$ with a hidden translation symmetry: moving one unit horizontally or vertically returns you to the same position. The symmetry group of this tiling — the deck transformations of the cover — is exactly $\mathbb{Z} \times \mathbb{Z}$, matching $\pi_1(T)$. [/example] [illustration:torus-universal-cover] [remark: Deck Transformations] The symmetry group of a universal cover is the group of deck transformations — homeomorphisms $\phi: \tilde{X} \to \tilde{X}$ such that $p \circ \phi = p$. Under the identification of $\tilde{X}$ with the space of based paths in $X$, the deck transformation corresponding to $[\gamma] \in \pi_1(X, x_0)$ acts by prepending $\gamma$: $[\alpha] \mapsto [\gamma \cdot \alpha]$. This gives the left action of $\pi_1(X, x_0)$ on $\tilde{X}$, which is the group-theoretic shadow of the monodromy action on fibres. [/remark] The existence theorem shows that the conditions on $X$ are sufficient. Necessity of semi-local simple connectivity follows from the argument preceding the definition: if $X$ has a universal cover, then every sufficiently small loop must be null-homotopic in $X$, which is exactly the semi-local simple connectivity condition. The condition is not as restrictive as it may sound: every manifold, every CW complex, and every "reasonable" space arising in practice is semi-locally simply connected. ## The Galois Correspondence We began the chapter asking what fundamental groups act on. We now have a complete answer: the fundamental group $\pi_1(X, x_0)$ acts on the fibre $p^{-1}(x_0)$ of any covering, and this action encodes the relationship between the covering and the subgroup $p_* \pi_1(\tilde{X}, \tilde{x}_0)$. The Galois correspondence makes this relationship into a precise bijection between covering spaces and subgroups. The analogy with Galois theory is not superficial. In field theory, one studies extensions $K/F$ by looking at the Galois group $\operatorname{Gal}(K/F)$ and the lattice of intermediate fields, each corresponding to a subgroup of $\operatorname{Gal}(K/F)$. Here, the role of $K$ is played by the universal cover $\tilde{X}$, the role of $F$ by $X$, and the role of $\operatorname{Gal}(K/F)$ by $\pi_1(X, x_0)$. Intermediate covering spaces between $\tilde{X}$ and $X$ correspond to intermediate subgroups. [definition: Based Covering Space Isomorphism] Two based covering spaces $p_1: (\tilde{X}_1, \tilde{x}_1) \to (X, x_0)$ and $p_2: (\tilde{X}_2, \tilde{x}_2) \to (X, x_0)$ are **isomorphic** if there is a homeomorphism $h: (\tilde{X}_1, \tilde{x}_1) \to (\tilde{X}_2, \tilde{x}_2)$ with $p_2 \circ h = p_1$. [/definition] The key technical tool for establishing the correspondence is a general existence criterion for lifts. [quotetheorem:1897] [citeproof:1897] The lifting criterion deserves attention: the condition $f_* \pi_1(Y, y_0) \leq p_* \pi_1(\tilde{X}, \tilde{x}_0)$ says that every loop in $Y$ that $f$ sends into $X$ must already be "accounted for" by the covering. When $Y$ is simply connected — as when $Y = I$ or $Y = I^2$ — the condition holds vacuously, recovering the path lifting and homotopy lifting lemmas as special cases. With the lifting criterion in hand, we can now prove the main theorem of the chapter. [quotetheorem:1898] [citeproof:1898] The correspondence between the number of sheets and the index of the subgroup is worth making explicit. If $H = p_* \pi_1(\tilde{X}, \tilde{x}_0)$, then the bijection $H \backslash \pi_1(X, x_0) \to p^{-1}(x_0)$ from the orbits-and-stabilisers theorem says exactly that the number of sheets equals $[\pi_1(X, x_0) : H]$. For the universal cover, $H = \{e\}$ and the index equals $|\pi_1(X, x_0)|$. For a degree-$n$ covering of $S^1$, the corresponding subgroup is $n\mathbb{Z} \leq \mathbb{Z}$, which has index $n$ — exactly the number of sheets. Passing from based to unbased covering spaces introduces an ambiguity. When we change the basepoint $\tilde{x}_0$ to another point $\tilde{x}_0'$ in the same fibre, the subgroup $p_* \pi_1(\tilde{X}, \tilde{x}_0)$ changes: if $\tilde{x}_0' = \tilde{x}_0 \cdot [\gamma]$, then the new subgroup is conjugate to the old one by $[\gamma]$. Different basepoints in the fibre are in bijection with elements of $\pi_1(X, x_0) / H$, and each choice conjugates $H$ by the corresponding element. [quotetheorem:1899] [citeproof:1899] The full Galois correspondence is summarised in the following dictionary: | Covering spaces of $X$ | Fundamental group $\pi_1(X, x_0)$ | |---|---| | Based covering spaces (up to based iso) | Subgroups $H \leq \pi_1(X, x_0)$ | | Unbased covering spaces (up to iso) | Conjugacy classes of subgroups | | Number of sheets | Index $[\pi_1(X, x_0) : H]$ | | Universal cover | Trivial subgroup $\{e\}$ | | $X$ itself | Whole group $\pi_1(X, x_0)$ | | Normal covering | Normal subgroup $H \trianglelefteq \pi_1(X, x_0)$ | A covering $p: \tilde{X} \to X$ is called normal (or regular, or Galois) when $p_* \pi_1(\tilde{X}, \tilde{x}_0)$ is a normal subgroup of $\pi_1(X, x_0)$. For such coverings, the deck transformation group is isomorphic to the quotient $\pi_1(X, x_0) / p_* \pi_1(\tilde{X}, \tilde{x}_0)$, and the deck transformations act freely and transitively on each fibre — a complete analogy with Galois extensions. [example: Normal Coverings of the Circle] For $X = S^1$ with $\pi_1(S^1, 1) = \mathbb{Z}$, every subgroup is normal (since $\mathbb{Z}$ is abelian). The subgroup $n\mathbb{Z}$ corresponds to the degree-$n$ covering $p_n: S^1 \to S^1$, $z \mapsto z^n$. The deck transformation group is $\mathbb{Z}/n\mathbb{Z}$, acting by rotations of the covering circle by $k/n$th of a full turn for $k = 0, \ldots, n-1$. The universal cover (corresponding to $\{0\} \leq \mathbb{Z}$) is $\mathbb{R} \to S^1$, and the deck transformations are translations by integers, forming a group isomorphic to $\mathbb{Z} \cong \pi_1(S^1, 1)$. [/example] The Galois correspondence is one of the most structurally satisfying results in algebraic topology. It transforms the question of classifying covering spaces — a topological problem — into the question of classifying subgroups of a group — an algebraic problem. The subsequent chapters develop homology, which provides invariants strong enough to distinguish spaces whose fundamental groups agree but whose higher topology differs. # 5. Some Group Theory The ultimate goal of algebraic topology is to translate topological problems into group-theoretic ones, where we have far more computational power. But this translation programme immediately raises a practical difficulty: the groups that arise — fundamental groups of cell complexes — are not abelian groups or matrix groups that we might recognise from earlier courses. They are described by generators and relations, and to work with them rigorously we need a careful theory of what such a description actually means. This chapter develops that theory from the ground up: free groups, group presentations, free products, and free products with amalgamation. These are precisely the algebraic ingredients needed for the Seifert–van Kampen theorem, which allows us to compute $\pi_1$ of spaces built by gluing. ## Free Groups and Their Presentations How do we make sense of an expression like $D_{2n} = \langle r, s \mid r^n = s^2 = e,\, srs = r^{-1} \rangle$? This notation, familiar from IA Groups, suggests that $D_{2n}$ is somehow built from two abstract symbols $r$ and $s$ subject to three constraints. But what is the ambient universe in which these symbols live before the relations are imposed? The answer is the *free group*, which we now define carefully. [definition: Alphabet and Words] Let $S = \{s_\alpha : \alpha \in \Lambda\}$ be a set, which we call an **alphabet**. Introduce a disjoint set of formal symbols $S^{-1} = \{s_\alpha^{-1} : \alpha \in \Lambda\}$, with $S \cap S^{-1} = \varnothing$. A **word** over $S$ is a finite sequence $x_1 x_2 \cdots x_n$ where each $x_i \in S \cup S^{-1}$; the case $n = 0$ gives the **empty word** $\varnothing$. We write $S^*$ for the set of all such words. [/definition] [example: Words on Two Generators] Take $S = \{a, b\}$. Then $S^*$ contains the empty word $\varnothing$, the single-letter words $a$, $b$, $a^{-1}$, $b^{-1}$, and longer words such as $aba^{-1}b^{-1}$ and $aa^{-1}aaaaabbbb$. For compactness, repeated letters are written with exponents: $aa^{-1}a^5b^4$ rather than $aa^{-1}aaaaabbbb$. [/example] The set $S^*$ of all words carries a natural multiplication (concatenation), but it is not yet a group: the word $aa^{-1}$ should be the same as the empty word, but as a sequence it is not. We need a reduction procedure. [definition: Elementary Reduction and Reduced Words] An **elementary reduction** of a word replaces a subword $s_\alpha s_\alpha^{-1}$ or $s_\alpha^{-1} s_\alpha$ by nothing (i.e., removes that adjacent pair). A word is **reduced** if it admits no elementary reduction — that is, no two consecutive letters are mutually inverse. [/definition] Since every elementary reduction strictly shortens the word, and words have finite length, any word can be reduced in finitely many steps. The question is whether different sequences of reductions always lead to the same reduced word. This is in fact true, but the proof is not immediate from the definition — we will establish it using topology. For now we provisionally define: [definition: Free Group] The **free group** on the set $S$, written $F(S)$, is the set of all reduced words over $S \cup S^{-1}$, equipped with: 1. **Multiplication**: concatenation followed by reduction to a reduced word. 2. **Identity**: the empty word $\varnothing$. 3. **Inverse**: the inverse of $x_1 x_2 \cdots x_n$ is $x_n^{-1} \cdots x_2^{-1} x_1^{-1}$, where $(s_\alpha^{-1})^{-1} := s_\alpha$. The elements of $S$ inside $F(S)$ are called the **generators** of $F(S)$. [/definition] The definition is provisional because multiplication might not be well-defined: different reduction sequences for the same word could, in principle, yield different results. We will resolve this rigorously in the next section using the universal cover of a graph. First, let us state the characteristic property of the free group that distinguishes it among all groups generated by $S$. [quotetheorem:1900] [citeproof:1900] This universal property is more than a convenience: it characterises $F(S)$ up to isomorphism. If two groups both satisfy the conclusion of this lemma for every target $G$, then taking $G$ to be each of them in turn and using the uniqueness clause forces an isomorphism between them. This means we can work with free groups purely through their universal property without worrying about the particulars of the word construction. Having defined free groups, we can now make precise the notation for groups defined by generators and relations. [definition: Presentation of a Group] Let $S$ be a set and $R \subset F(S)$ a subset of elements of the free group. The **normal closure** of $R$ in $F(S)$, written $\langle\langle R \rangle\rangle$, is the smallest normal subgroup of $F(S)$ containing $R$; explicitly, \begin{align*} \langle\langle R \rangle\rangle = \left\{ \prod_{i=1}^n g_i r_i g_i^{-1} : n \in \mathbb{N},\, r_i \in R,\, g_i \in F(S) \right\}. \end{align*} The quotient $\langle S \mid R \rangle := F(S) / \langle\langle R \rangle\rangle$ is called the group **presented** by generators $S$ and relations $R$. [/definition] The definition makes precise what it means to impose relations: we take the largest possible group on $S$ (the free group), and then force exactly the elements of $R$ to become the identity by passing to the quotient. No other relations are imposed beyond those forced by normality and the group axioms. [example: The Canonical Presentation] Every group has a presentation. Given any group $G$, treat its underlying set as an alphabet and form $F(G)$. The identity map $G \to G$ is a set map, so the universal property of $F(G)$ yields a surjective homomorphism $f: F(G) \to G$. Setting $R = \ker f$, the first isomorphism theorem gives $G \cong F(G)/R = \langle G \mid R \rangle$. This presentation is impractical — even $\mathbb{Z}/2$ would be presented with infinitely many generators — but it confirms that every group can be described this way. [/example] [example: Recognising Familiar Groups] The presentation $\langle a, b \mid b \rangle$ forces $b = e$ in the quotient, leaving a free group on one generator $a$. A free group on one generator is $\mathbb{Z}$ (the generator corresponds to $1$, its inverse to $-1$, and powers give all integers). So $\langle a, b \mid b \rangle \cong \mathbb{Z}$. With more work, one can verify that $\langle a, b \mid ab^{-3}, ba^{-2} \rangle \cong \mathbb{Z}/5$: the two relations give $a = b^3$ and $b = a^2$, so $a = (a^2)^3 = a^6$, hence $a^5 = e$ and $a$ generates $\mathbb{Z}/5$. [/example] The presentation $\langle a, b \mid b \rangle \cong \mathbb{Z}$ example also illustrates the universal property of presentations: given any group $G$ and a map $\phi: S \to G$ such that each relation $r \in R$ maps to the identity in $G$, there is a unique homomorphism $\langle S \mid R \rangle \to G$ extending $\phi$. This says that $\langle S \mid R \rangle$ is the largest group generated by $S$ satisfying the given relations and nothing more. ## The Topology of Free Groups Multiplication in $F(S)$ was defined as concatenation followed by reduction, and we noted that this might not be well-defined: two different reduction sequences for the same word could in principle terminate at different reduced words. We now resolve this by constructing $F(S)$ topologically, which simultaneously gives a beautiful and concrete picture of what the free group looks like. The problem of well-definedness asks whether every word reduces to a unique reduced form. The topological approach sidesteps this by identifying reduced words with paths in a tree, where paths are inherently unique. Let $S$ be any set. For concreteness, take $S = \{a, b\}$. Define a topological space $X$ as a **rose with $|S|$ petals**: one vertex $x_0$ (a 0-cell) and one loop (1-cell) for each generator $s \in S$. For $S = \{a, b\}$ this is a figure-eight: two circles sharing the single point $x_0$, with the loop labeled $a$ on the left and $b$ on the right. [illustration:rose-two-petals] The fundamental group $\pi_1(X, x_0)$ is what we want to identify with $F(S)$. To do this, we study the universal cover $\widetilde{X}$. Since $X$ is a 1-complex (a graph), $\widetilde{X}$ is also a graph. Being the universal cover, $\widetilde{X}$ is connected and simply connected, hence a tree. Since the covering map $p: \widetilde{X} \to X$ is a local homeomorphism, every vertex $\tilde{x} \in \widetilde{X}$ has the same local structure as $x_0 \in X$: four half-edges emanate from it, one for each of $a$, $a^{-1}$, $b$, $b^{-1}$ (an incoming and outgoing edge for each generator). This means $\widetilde{X}$ is the infinite 4-regular tree, with each vertex of degree 4. The key observations are: 1. Every word $w \in S^*$ describes a unique edge-path in $\widetilde{X}$ starting at the chosen lift $\tilde{x}_0$ of $x_0$: the letter $s \in S$ tells us to traverse the unique $s$-labeled edge leaving the current vertex forward, while $s^{-1}$ tells us to traverse it backward. 2. An edge-path fails to be locally injective precisely when two consecutive edges are the same edge traversed in opposite directions — which corresponds exactly to an elementary reduction in the word. 3. Since $\widetilde{X}$ is a tree, any two vertices are connected by a unique locally injective (reduced) edge-path. 4. The fibre $p^{-1}(x_0)$ is therefore in bijection with the set of reduced words in $F(S)$: each vertex $\tilde{x}$ above $x_0$ corresponds to the unique reduced word $w$ such that the reduced edge-path from $\tilde{x}_0$ ends at $\tilde{x}$. Separately, the theory of covering spaces gives a bijection between $p^{-1}(x_0)$ and $\pi_1(X, x_0)$: a homotopy class $[\gamma] \in \pi_1(X, x_0)$ corresponds to $\tilde{x}_0 \cdot [\gamma]$, the endpoint of the unique lift of $\gamma$ starting at $\tilde{x}_0$. Combining the two bijections: [quotetheorem:1902] [citeproof:1902] This result has two important consequences. First, it proves that multiplication in $F(S)$ is well-defined: since $F(S) \cong \pi_1(X, x_0)$, and $\pi_1$ is a genuine group with well-defined multiplication, so is $F(S)$. Second, it gives us geometric intuition for free groups — an element of $F(S)$ is literally a loop in a graph, and the product is concatenation of loops. [illustration:universal-cover-rose-tree] ## Free Products and Amalgamation We can now compute $\pi_1(S^1) \cong \mathbb{Z}$. The next challenge is to compute the fundamental groups of spaces assembled from simpler pieces. If $X = A \cup B$, how does $\pi_1(X)$ relate to $\pi_1(A)$ and $\pi_1(B)$? Before we can state the answer (the Seifert–van Kampen theorem), we need the algebraic construction that models group-level gluing. The most basic version: if $A$ and $B$ share only a single simply-connected point, the fundamental group of $A \cup B$ should combine $\pi_1(A)$ and $\pi_1(B)$ with no interaction. The algebraic counterpart is the free product. [definition: Free Product] Let $G_1 = \langle S_1 \mid R_1 \rangle$ and $G_2 = \langle S_2 \mid R_2 \rangle$ be groups, with $S_1 \cap S_2 = \varnothing$. The **free product** $G_1 * G_2$ is the group \begin{align*} G_1 * G_2 = \langle S_1 \cup S_2 \mid R_1 \cup R_2 \rangle. \end{align*} There are natural homomorphisms $j_i: G_i \to G_1 * G_2$ sending each generator of $G_i$ to the corresponding generator of $G_1 * G_2$. [/definition] A group can have many presentations, so defining $G_1 * G_2$ by a particular choice of presentations $\langle S_i \mid R_i \rangle$ requires justification. The universal property provides it. [quotetheorem:1903] [citeproof:1903] The universal property shows that $G_1 * G_2$ is determined up to isomorphism by the property stated, independent of the choice of presentations. Concretely, elements of $G_1 * G_2$ are alternating products $g_1 h_1 g_2 h_2 \cdots$ with $g_i \in G_1 \setminus \{e\}$ and $h_i \in G_2 \setminus \{e\}$, which is the group-theoretic analogue of concatenating reduced words. The free product models gluing spaces along a simply-connected intersection, but most interesting topological gluings involve a nontrivial overlap. When $X = A \cup B$ with $A \cap B$ connected but not simply connected, the fundamental group of the overlap maps into both $\pi_1(A)$ and $\pi_1(B)$, and these images must be identified. The algebraic structure that captures this is the free product with amalgamation. [definition: Free Product with Amalgamation] Let $G_1$, $G_2$, and $H$ be groups with homomorphisms $i_1: H \to G_1$ and $i_2: H \to G_2$. The **free product with amalgamation** (or **amalgamated free product**) is \begin{align*} G_1 *_H G_2 = G_1 * G_2 / \langle\langle \{ (j_1 \circ i_1(h))^{-1}(j_2 \circ i_2(h)) : h \in H \} \rangle\rangle, \end{align*} where $j_k: G_k \to G_1 * G_2$ are the canonical inclusions. [/definition] The construction imposes exactly the relations that force $j_1(i_1(h)) = j_2(i_2(h))$ for all $h \in H$ — that is, elements coming from $H$ via $i_1$ and elements coming via $i_2$ are identified in the quotient. No other relations are imposed beyond these. To see why this is the right construction, consider the case $H = \{e\}$: both $i_k$ are trivial, the relations become $j_1(e)^{-1} j_2(e) = e$, which is already true, so $G_1 *_{\{e\}} G_2 = G_1 * G_2$. The free product is the special case of amalgamation over the trivial group. [quotetheorem:1904] The condition $\phi_1 \circ i_1 = \phi_2 \circ i_2$ is necessary: the amalgamation identifies the two images of $H$, so any homomorphism out of $G_1 *_H G_2$ must send both to the same subgroup of $K$. The universal property says this is the only obstruction — if the two maps agree on $H$, they combine uniquely into a map from the amalgamated product. This framework sets up exactly the algebraic language needed for the Seifert–van Kampen theorem. When a space $X = A \cup B$ is expressed as a union with open subsets $A$, $B$, and $A \cap B$ all path-connected, the theorem identifies $\pi_1(X)$ with the free product $\pi_1(A) *_{\pi_1(A \cap B)} \pi_1(B)$, where the amalgamation is over the two maps $\pi_1(A \cap B) \to \pi_1(A)$ and $\pi_1(A \cap B) \to \pi_1(B)$ induced by the inclusions. In the next chapter we prove this theorem and deploy it to compute fundamental groups of surfaces, knot complements, and other spaces that would be out of reach by direct inspection. # 6. The Seifert–van Kampen Theorem This chapter addresses a central question in the computation of fundamental groups: if a space $X$ is assembled from two simpler pieces $A$ and $B$, can we reconstruct $\pi_1(X)$ from $\pi_1(A)$, $\pi_1(B)$, and the way $A$ and $B$ overlap? The Seifert–van Kampen theorem answers this question precisely, expressing $\pi_1(X)$ as a free product with amalgamation. We then apply this theorem systematically — first to recover familiar groups like $\{e\}$ for higher spheres, then to analyse surfaces, and finally to see how cell attachments sculpt the fundamental group one cell at a time. ## The Statement and Its Algebraic Backbone Before writing down the theorem, we need the algebraic object that will appear in its conclusion. [definition: Free Product with Amalgamation] Let $G$, $H$, and $K$ be groups, with group homomorphisms $\phi: K \to G$ and $\psi: K \to H$. The **free product with amalgamation** $G *_K H$ is the quotient of the free product $G * H$ by the normal closure of the set $\{\phi(k)\psi(k)^{-1} : k \in K\}$. Equivalently, it is the unique group (up to isomorphism) satisfying the following universal property: for every group $Q$ and homomorphisms $f: G \to Q$, $g: H \to Q$ such that $f \circ \phi = g \circ \psi$, there exists a unique homomorphism $h: G *_K H \to Q$ with $h \circ i_G = f$ and $h \circ i_H = g$, where $i_G: G \to G *_K H$ and $i_H: H \to G *_K H$ are the canonical inclusions. [/definition] The universal property is the key structural fact to keep in mind: whenever two homomorphisms from $G$ and from $H$ agree on the amalgamated subgroup $K$, they combine into a unique homomorphism out of $G *_K H$. This is precisely the property that will match the topology of the situation. Now consider a space $X = A \cup B$ where $A$, $B$, and $A \cap B$ are all path-connected. Pick a basepoint $x_0 \in A \cap B$. The inclusion maps \begin{align*} A \cap B \hookrightarrow A, \qquad A \cap B \hookrightarrow B, \qquad A \hookrightarrow X, \qquad B \hookrightarrow X \end{align*} induce group homomorphisms between fundamental groups, fitting into a commutative square \begin{align*} \pi_1(A \cap B, x_0) &\longrightarrow \pi_1(B, x_0) \\ \pi_1(A \cap B, x_0) &\longrightarrow \pi_1(A, x_0) \longrightarrow \pi_1(X, x_0). \end{align*} The two compositions $\pi_1(A \cap B, x_0) \to \pi_1(A, x_0) \to \pi_1(X, x_0)$ and $\pi_1(A \cap B, x_0) \to \pi_1(B, x_0) \to \pi_1(X, x_0)$ agree (both are just the map induced by inclusion $A \cap B \hookrightarrow X$). By the universal property of the free product with amalgamation, there is therefore a unique homomorphism \begin{align*} \Phi: \pi_1(A, x_0) *_{\pi_1(A \cap B, x_0)} \pi_1(B, x_0) \longrightarrow \pi_1(X, x_0). \end{align*} The Seifert–van Kampen theorem says that, under mild hypotheses, $\Phi$ is an isomorphism. [quotetheorem:1905] [citeproof:1905] The surjectivity argument is the more geometric half: every loop in $X$ can be subdivided into arcs each lying in $A$ or $B$, and then reconnected via $A \cap B$. The injectivity argument is the harder combinatorial half, where one must show that any algebraic cancellation visible in $\pi_1(X)$ was already present in the amalgamated free product. The hypothesis that $A$, $B$, and $A \cap B$ are open and path-connected cannot be dropped without replacement: if $A \cap B$ is disconnected, loops in the intersection can be pushed into different path-components, yielding a free product rather than an amalgamation, and the theorem fails. We address a refinement for closed subspaces in a later section. [illustration:seifert-van-kampen-cover] ## First Applications: Spheres and Projective Spaces The first payoff of the Seifert–van Kampen theorem is a clean calculation of the fundamental group of the $n$-sphere for $n \geq 2$. [example: Fundamental Group of S^n for n at least 2] Fix $n \geq 2$ and let $S^n = \{v \in \mathbb{R}^{n+1} : |v| = 1\}$. We want to compute $\pi_1(S^n)$. Write $S^n = A \cup B$ where $A = S^n \setminus \{n\}$ (the sphere minus the north pole $n = e_1$) and $B = S^n \setminus \{s\}$ (the sphere minus the south pole $s = -e_1$). By stereographic projection from the respective poles, $A \cong \mathbb{R}^n$ and $B \cong \mathbb{R}^n$, so both are contractible and their fundamental groups are $\{e\}$. The intersection $A \cap B = S^n \setminus \{n, s\}$ is homeomorphic to $S^{n-1} \times (-1, 1)$: radial projection maps $A \cap B$ bijectively onto the cylinder over the equatorial sphere, and the projection in the other direction is its inverse (both maps are continuous). Since $(-1, 1)$ is contractible, $A \cap B \simeq S^{n-1}$. For $n \geq 2$, the sphere $S^{n-1}$ is path-connected (it requires $n - 1 \geq 1$, i.e.\ $n \geq 2$), so the hypotheses of the Seifert–van Kampen theorem are satisfied. We obtain \begin{align*} \pi_1(S^n) \cong \pi_1(A) *_{\pi_1(A \cap B)} \pi_1(B) \cong \{e\} *_{\pi_1(S^{n-1})} \{e\}. \end{align*} The amalgamation of two copies of $\{e\}$ is $\{e\}$ regardless of the subgroup being amalgamated, since the entire amalgamated product is a quotient of $\{e\} * \{e\} = \{e\}$. Therefore $\pi_1(S^n) \cong \{e\}$ for all $n \geq 2$. [/example] This result deserves a moment's reflection. Spheres $S^n$ for $n \geq 2$ are simply connected, yet they are emphatically not contractible (this will be confirmed later when we compute their higher homology groups). So simple connectivity does not imply contractibility — the two notions coincide for convex subsets of $\mathbb{R}^n$, but not in general. The result also explains why the direct approach of "find a point not on the loop and project stereographically" is subtle: space-filling curves make it impossible to guarantee such a point exists for an arbitrary continuous loop. The Seifert–van Kampen argument sidesteps this issue entirely. One might wonder why the case $n = 1$ is excluded. When $n = 1$ we have $S^0 = \{-1, 1\}$, which has two path-components. The intersection $A \cap B = S^1 \setminus \{n, s\}$ is therefore disconnected, violating the path-connectivity hypothesis. Indeed, $\pi_1(S^1) \cong \mathbb{Z}$, not $\{e\}$. [example: Fundamental Group of RP^n] Since $S^n$ is simply connected for $n \geq 2$, we can determine $\pi_1(\mathbb{RP}^n)$ using covering space theory. Recall that $\mathbb{RP}^n = S^n / \{\pm \mathrm{id}\}$, and the quotient map $p: S^n \to \mathbb{RP}^n$ is a 2-sheeted covering. Because $S^n$ is simply connected (for $n \geq 2$), it serves as the universal cover of $\mathbb{RP}^n$. For any $x_0 \in \mathbb{RP}^n$, covering space theory gives a bijection $\pi_1(\mathbb{RP}^n, x_0) \leftrightarrow p^{-1}(x_0)$. The fibre $p^{-1}(x_0)$ consists of exactly two antipodal points $\{v, -v\} \subset S^n$. Therefore $|\pi_1(\mathbb{RP}^n, x_0)| = 2$, so $\pi_1(\mathbb{RP}^n) \cong \mathbb{Z}/2$. [/example] This short argument combines two chapters of theory. The Seifert–van Kampen theorem established that $S^n$ is simply connected, and the general correspondence between covering spaces and subgroups of $\pi_1$ then read off the fundamental group of $\mathbb{RP}^n$ from the size of a fibre. The same argument extends to any regular covering: if $p: \tilde{X} \to X$ is a covering with $\tilde{X}$ simply connected and every fibre has exactly $m$ elements, then $\pi_1(X) \cong \mathbb{Z}/m$ (when the deck group is cyclic) or, more generally, the group of deck transformations. ## Wedge Sums and Free Products The wedge sum is the based analogue of the disjoint union. Given based spaces $(X, x_0)$ and $(Y, y_0)$, the **wedge sum** $X \vee Y$ is the quotient of the disjoint union $X \amalg Y$ obtained by identifying $x_0 \sim y_0$. The two basepoints become a single point in $X \vee Y$. [example: Fundamental Group of the Figure Eight] Let $X = S^1 \vee S^1$, the wedge of two circles sharing a single basepoint $x_0$. We wish to compute $\pi_1(S^1 \vee S^1, x_0)$. To apply Seifert–van Kampen, we need open sets. Let $A$ be a small open neighbourhood of the first circle — concretely, take the first circle together with a small open arc of the second circle containing $x_0$. Similarly, let $B$ be the second circle together with a small arc of the first circle. Each of $A$ and $B$ deformation retracts onto its respective circle, so $\pi_1(A, x_0) \cong \mathbb{Z}$ and $\pi_1(B, x_0) \cong \mathbb{Z}$. The intersection $A \cap B$ is a small cross-shaped neighbourhood of $x_0$, which deformation retracts to the single point $\{x_0\}$. Therefore $\pi_1(A \cap B, x_0) = \{e\}$. Applying Seifert–van Kampen: \begin{align*} \pi_1(S^1 \vee S^1, x_0) \cong \mathbb{Z} *_{\{e\}} \mathbb{Z} \cong \mathbb{Z} * \mathbb{Z} \cong F_2, \end{align*} where $F_2$ denotes the free group on two generators. When the amalgamated subgroup is $\{e\}$, the amalgamated free product reduces to the ordinary free product. [/example] The figure eight having fundamental group $F_2 = \langle a, b \rangle$ is a landmark result. It shows that fundamental groups need not be abelian — the free group $F_2$ contains elements like $aba^{-1}b^{-1}$ that are not the identity. The corresponding covering spaces of $S^1 \vee S^1$ are graphs on which $F_2$ acts, and studying them is a powerful approach to the combinatorial group theory of free groups. The calculation for the figure eight generalises to arbitrary wedge sums. As long as the basepoints $x_0 \in X$ and $y_0 \in Y$ are neighbourhood deformation retracts of their respective spaces (which holds whenever $X$ and $Y$ are cell complexes), one has \begin{align*} \pi_1(X \vee Y) \cong \pi_1(X) * \pi_1(Y). \end{align*} The proof is essentially the same: $A \cap B$ contracts to the basepoint, collapsing the amalgamated subgroup to $\{e\}$ and turning the amalgamation into a free product. ### Covering Spaces of the Figure Eight Since $\pi_1(S^1 \vee S^1) \cong F_2 = \langle a, b \rangle$, every subgroup of $F_2$ corresponds to a connected covering space of $S^1 \vee S^1$. We can construct these coverings explicitly using **Schreier graphs**: the vertices are cosets of the subgroup, and we draw directed edges labelled $a$ and $b$ according to the right-multiplication action. [example: A Three-Sheeted Covering from Z/3] Let $\phi: F_2 \to \mathbb{Z}/3$ be defined by $\phi(a) = 1$ and $\phi(b) = 1$. Since $\mathbb{Z}/3$ is presented freely by specifying where generators go, $\phi$ is a well-defined surjective homomorphism. The kernel $K = \ker \phi$ is a subgroup of $F_2$ of index $|\mathbb{Z}/3| = 3$ (by the first isomorphism theorem), so the corresponding covering $\tilde{X} \to S^1 \vee S^1$ has three sheets, meaning three preimages over the basepoint $x_0$. Label these vertices $0, 1, 2 \in \mathbb{Z}/3$. The lifted edge labelled $a$ starting at vertex $i$ goes to vertex $i + 1 \pmod{3}$, because $\phi(a) = 1$ shifts the coset. The same holds for the $b$-edges. The resulting covering space is a graph with three vertices and six directed edges (three $a$-edges and three $b$-edges), each set forming a directed $3$-cycle. Concretely, both $a$ and $b$ act as the cyclic permutation $0 \mapsto 1 \mapsto 2 \mapsto 0$ on the vertex set. One can verify that this covering space is a well-defined graph on which the projection to $S^1 \vee S^1$ is indeed a covering map: every preimage of $x_0$ has exactly one incoming and one outgoing $a$-edge and one incoming and one outgoing $b$-edge. [/example] This construction is a general recipe. Given any finite group $G$ and elements $\alpha, \beta \in G$, defining $\phi: F_2 \to G$ by $a \mapsto \alpha$, $b \mapsto \beta$ produces a covering of $S^1 \vee S^1$ with $|G|$ sheets. Non-isomorphic subgroups of $F_2$ — even of the same index — yield non-homeomorphic covering spaces, so the covering theory of the figure eight encodes a rich family of combinatorial structures. ## Attaching Cells and Killing Loops We now return to cell complexes. Recall that given a space $X$ and a map $f: S^{n-1} \to X$, the space $X \cup_f D^n$ is obtained by attaching an $n$-cell to $X$ along $f$. The Seifert–van Kampen theorem predicts how this changes $\pi_1(X)$. [quotetheorem:1906] [citeproof:1906] The critical observation is that the intersection $A \cap B \simeq S^{n-1}$ contributes nothing algebraically when $n \geq 3$, because $\pi_1(S^{n-1}) = \{e\}$ for $n - 1 \geq 2$. Attaching a cell of dimension three or higher is topologically significant (it changes the higher homotopy groups) but leaves $\pi_1$ unchanged. The case $n = 2$ is where the interesting algebra happens. [quotetheorem:1908] [citeproof:1908] Geometrically, attaching a 2-disk along a loop $f$ "fills in" that loop, forcing it to be null-homotopic in the new space. The normal closure $\langle\langle [f] \rangle\rangle$ appears because all conjugates of $[f]$ also become the identity: conjugating $[f]$ by another loop $[\gamma]$ corresponds to sliding the basepoint of $f$ around, and all such variations get killed when $f$ bounds a disk. The theorem does not say that attaching the disk makes $[f]$ the only element killed — the normal closure can be much larger than the cyclic group generated by $[f]$. Combining both theorems, we obtain a clean summary: \begin{align*} \pi_1\!\left(X \cup_f D^n\right) \cong \begin{cases} \pi_1(X) & n \geq 3, \\ \pi_1(X) / \langle\langle [f] \rangle\rangle & n = 2. \end{cases} \end{align*} This is the engine that drives the computation of fundamental groups of cell complexes. ## Fundamental Groups of Cell Complexes The cell-attachment theorems give a recursive algorithm for computing $\pi_1$ of any finite cell complex. Start from the 0-skeleton (a discrete set of points, with fundamental group equal to $\{e\}$), attach 1-cells to build $X^{(1)}$, and attach 2-cells to build $X^{(2)}$. Higher-dimensional cells then leave $\pi_1$ untouched. [example: Fundamental Group of the Torus] The torus $T^2$ admits a cell complex structure with one 0-cell, two 1-cells $a$ and $b$, and one 2-cell. After identifying, the 1-skeleton $X^{(1)}$ is the wedge $S^1 \vee S^1$, so $\pi_1(X^{(1)}) \cong F_2 = \langle a, b \rangle$. The 2-cell is a square attached by the word $aba^{-1}b^{-1}$: going around the boundary of the square traces the loop $a$, then $b$, then $a^{-1}$, then $b^{-1}$. Applying the attaching theorem: \begin{align*} \pi_1(T^2) \cong F_2 / \langle\langle aba^{-1}b^{-1} \rangle\rangle = \langle a, b \mid aba^{-1}b^{-1} \rangle \cong \mathbb{Z}^2. \end{align*} The relation $aba^{-1}b^{-1} = e$ forces $ab = ba$, making $a$ and $b$ commute. The abelian group freely generated by two commuting generators is $\mathbb{Z}^2$. [/example] This matches what we know from the product structure $T^2 = S^1 \times S^1$ and the formula $\pi_1(X \times Y) \cong \pi_1(X) \times \pi_1(Y)$. The cell complex computation gives an independent confirmation, and more importantly it generalises to surfaces where a product structure is not available. The procedure can be reversed: given any group presentation $\langle S \mid R \rangle$, we can build a 2-dimensional cell complex realising it. [quotetheorem:1909] [citeproof:1909] Since every group — not just finitely presented ones — has some presentation, this shows that every group arises as the fundamental group of some topological space. The finite case is most natural, but the construction works in full generality by allowing infinite cell complexes. This is a remarkable realisation theorem: abstract group theory and topology are in tight correspondence at the level of $\pi_1$. ## A Refinement for Closed Subspaces In practice, the openness hypothesis of the original Seifert–van Kampen theorem is inconvenient. When $X$ is decomposed as a union of closed subsets — as often happens with subcomplexes — one must artificially thicken the pieces to make them open. A cleaner formulation avoids this. [definition: Neighbourhood Deformation Retract] A subspace $A \subset X$ is a **neighbourhood deformation retract** if there exists an open set $U$ with $A \subset U \subset X$ such that $A$ is a strong deformation retract of $U$ — that is, there is a retraction $r: U \to A$ and a homotopy $H: U \times [0,1] \to U$ with $H(-,0) = \mathrm{id}_U$, $H(-,1) = r$, and $H(a,t) = a$ for all $a \in A$ and $t \in [0,1]$. [/definition] The neighbourhood deformation retract condition is automatic in the setting of cell complexes: any subcomplex of a CW complex is a neighbourhood deformation retract of the ambient complex. This makes the following version of the theorem considerably easier to apply. [quotetheorem:1910] [citeproof:1910] This refinement shows that the earlier calculations were not deficient — they simply carried out the thickening step implicitly. The figure eight example is now immediate without any artificial enlargement: taking $A$ and $B$ to be the two closed circles and observing that $A \cap B = \{x_0\}$ is a neighbourhood deformation retract of each circle (since circles are manifolds and points are always NDRs in manifolds), we recover $\pi_1(S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z}$ directly. ## The Fundamental Groups of All Compact Surfaces The most substantial application in this chapter is the computation of $\pi_1$ for every compact surface. We first recall the classification theorem, whose full proof we will not rehearse here (its proof belongs to a full course on geometric topology). [quotetheorem:1911] Each surface in this list admits a polygon model: $\Sigma_g$ is a $4g$-gon with edges identified in pairs, and $E_n$ is a $2n$-gon with edges identified. The polygon model is precisely a cell complex structure, which makes the fundamental group computable. [example: Fundamental Group of the Genus-g Surface] The orientable surface $\Sigma_g$ of genus $g$ is obtained from a $4g$-gon by identifying edges according to the word \begin{align*} a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1}. \end{align*} All $4g$ corners of the polygon get identified to a single vertex (the basepoint $x_0$), and all $2g$ pairs of edges produce $2g$ distinct 1-cells $a_1, b_1, \ldots, a_g, b_g$ in the resulting cell complex. The 1-skeleton is therefore a wedge of $2g$ circles, with $\pi_1(X^{(1)}) \cong F_{2g} = \langle a_1, b_1, \ldots, a_g, b_g \rangle$. Attaching the single 2-cell along the word above gives: \begin{align*} \pi_1(\Sigma_g) = \langle a_1, b_1, \ldots, a_g, b_g \mid a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1} \rangle. \end{align*} To see this concretely for $g = 2$: the octagon gives $\pi_1(\Sigma_2) = \langle a_1, b_1, a_2, b_2 \mid a_1 b_1 a_1^{-1} b_1^{-1} a_2 b_2 a_2^{-1} b_2^{-1} \rangle$. One can visualise $\Sigma_2$ as two tori glued along a disk removed from each: cutting the octagon along a diagonal produces two squares, each giving a torus with one hole, and regluing along the boundary circle of the holes reconstructs $\Sigma_2$. [/example] These groups distinguish the surfaces from one another. Although it is not immediate from staring at the presentations that $\pi_1(\Sigma_g) \ncong \pi_1(\Sigma_{g'})$ for $g \neq g'$, a simple invariant settles the question: the **abelianisation** $\pi_1(\Sigma_g)^{\mathrm{ab}}$. Taking the quotient of $\pi_1(\Sigma_g)$ by its commutator subgroup sets all generators commuting, so all the relation $a_1 b_1 a_1^{-1} b_1^{-1} \cdots = e$ becomes automatic. The abelianisation is $\mathbb{Z}^{2g}$. Since $\mathbb{Z}^{2g} \ncong \mathbb{Z}^{2g'}$ when $g \neq g'$, the surfaces $\Sigma_g$ are pairwise non-homeomorphic — and in fact not even homotopy equivalent. The abelianisation $\pi_1(\Sigma_g)^{\mathrm{ab}} \cong \mathbb{Z}^{2g}$ is the first homology group $H_1(\Sigma_g)$, a preview of the homology theory that occupies the remaining chapters of this course. The integer $2g$ is encoded in the Euler characteristic $\chi(\Sigma_g) = 2 - 2g$, which can be computed directly from any cell decomposition as (vertices) $-$ (edges) $+$ (faces). This interplay between $\pi_1$, $H_1$, and $\chi$ is a recurring theme in the chapters ahead. # 7. Simplicial Complexes Having built up the fundamental group $\pi_1$ in considerable depth, the course now pivots to a different family of invariants — the homology groups $H_n(X)$. The fundamental group is powerful but difficult: computing $\pi_1$ requires understanding group presentations, and there is no algorithm to determine whether an arbitrary group presentation defines the trivial group. Homology replaces group-theoretic algebra with linear algebra, which is both computationally tractable and conceptually cleaner. This chapter develops the combinatorial scaffolding — simplicial complexes and the simplicial approximation theorem — on which the homology machine will be built. ## Why Homology Replaces Higher Homotopy Groups The pattern of the course so far suggests a natural programme: to distinguish $\mathbb{R}^m$ from $\mathbb{R}^n$, use the homotopy group $\pi_{n-1}$ of the punctured space $\mathbb{R}^n \setminus \{0\} \simeq S^{n-1}$. We already saw $\pi_0$ handle the case $m = 1$, and $\pi_1(S^1) \cong \mathbb{Z}$ handles $m = 2$. The natural next step is to define higher homotopy groups $\pi_n(X)$ for each $n$. The problem is severity: $\pi_1$ already consumed several weeks of work and required nontrivial computation. Higher homotopy groups are far harder, both to define and to compute. They will only be treated in Part III courses. The deeper reason to abandon this path is structural: there are too many groups, and the tools for distinguishing them are coarse. Even with a complete presentation of a group, no algorithm can decide in general whether that presentation describes the group $= \{e\}$. [motivation] ### From groups to linear algebra The replacement idea is to work with objects closer to linear algebra. Instead of assigning a group $\pi_n(X)$ to $X$, we will assign abelian groups $H_n(X)$ — the homology groups — which behave more like vector spaces and are computable by matrix methods. The passage from $\pi_1$ to $H_1$ captures some of the same topological information but in a form that computers can handle. Linear algebra is not merely easier in practice; it has a complete structural theory (rank, the structure theorem for finitely generated abelian groups) that lets us read off invariants immediately from a matrix. [/motivation] ## Simplicial Complexes The challenge in defining homology is: homology of *what*, exactly? We need the space to have enough combinatorial structure that we can define chain groups and boundary maps. The solution is to restrict attention to spaces that can be decomposed into elementary building blocks — simplices — assembled according to strict rules. Such a space is called a simplicial complex. ### Affine Independence and Simplices Before defining simplices, we need a notion of "general position" for a finite collection of points. In ordinary linear algebra, vectors are independent if no one of them is a linear combination of the others. Affine independence is the corresponding notion that does not privilege any particular origin. [definition: Affine Independence] A finite set of points $\{a_1, \ldots, a_n\} \subset \mathbb{R}^m$ is **affinely independent** if \begin{align*} \sum_{i=1}^n t_i a_i = 0 \text{ and } \sum_{i=1}^n t_i = 0 \implies t_i = 0 \text{ for all } i. \end{align*} [/definition] The definition can be restated in a more transparent way via the following lemma, which shows that affine independence is exactly linear independence after choosing one point as an origin. [quotetheorem:1912] [citeproof:1912] The upshot is that $n+1$ affinely independent points span an "$n$-dimensional thing" — more precisely, an $n$-simplex. This is precisely what we define next. [definition: $n$-Simplex] An **$n$-simplex** is the convex hull of $n+1$ affinely independent points $a_0, a_1, \ldots, a_n \in \mathbb{R}^m$: \begin{align*} \sigma = \langle a_0, \ldots, a_n \rangle = \left\{ \sum_{i=0}^n t_i a_i \;\Big|\; \sum_{i=0}^n t_i = 1,\; t_i \geq 0 \right\}. \end{align*} The points $a_0, \ldots, a_n$ are the **vertices** of $\sigma$, and the tuple $(t_0, \ldots, t_n)$ with $\sum t_i = 1$ and $t_i \geq 0$ are the **barycentric coordinates** of the point $\sum t_i a_i$. [/definition] The dimension zero through three cases are the building blocks one meets first: a $0$-simplex is a single point, a $1$-simplex is a line segment, a $2$-simplex is a solid triangle (the filled region), and a $3$-simplex is a solid tetrahedron. The barycentric coordinate system is the key feature: every point inside the simplex has a *unique* expression $\sum t_i a_i$ with $\sum t_i = 1$ and $t_i \geq 0$, because affine independence guarantees no relations among the vertices. [illustration:standard-simplices] A simplex has its own notion of faces, which captures the combinatorial boundary structure. [definition: Face, Boundary and Interior] A **face** of a simplex $\sigma = \langle a_0, \ldots, a_n \rangle$ is any subsimplex spanned by a subset of the vertices. The **boundary** $\partial \sigma$ is the union of all proper faces (i.e., faces strictly smaller than $\sigma$). The **interior** $\mathring{\sigma}$ is $\sigma \setminus \partial \sigma$, the complement of the boundary inside $\sigma$. We write $\tau \leq \sigma$ when $\tau$ is a face of $\sigma$. [/definition] [remark: Simplex Interior vs Topological Interior] The interior $\mathring{\sigma}$ defined here is the complement of the union of proper faces inside $\sigma$. This is *not* the same as the topological interior of $\sigma$ as a subset of $\mathbb{R}^m$: a $1$-simplex (a line segment) has empty topological interior as a subset of $\mathbb{R}^2$, but its simplicial interior consists of all points strictly between the two endpoints. The simplicial interior of a vertex is just the vertex itself. [/remark] The standard $n$-simplex is spanned by the standard basis vectors $e_0, e_1, \ldots, e_n$ in $\mathbb{R}^{n+1}$. For $n = 2$, this is the triangle in $\mathbb{R}^3$ with vertices at $e_0 = (1,0,0)$, $e_1 = (0,1,0)$, and $e_2 = (0,0,1)$, lying in the plane $t_0 + t_1 + t_2 = 1$. ### Gluing Simplices into Complexes A single simplex is too simple to capture interesting topology. The next step is to specify how simplices may be assembled into larger spaces. The rules of assembly must ensure that simplices meet along entire faces — not at interior points, not along partial edges — and that every face of a simplex in the complex is itself in the complex. [definition: Geometric Simplicial Complex] A **geometric simplicial complex** is a finite collection $K$ of simplices in $\mathbb{R}^m$ satisfying: 1. If $\sigma \in K$ and $\tau \leq \sigma$, then $\tau \in K$. 2. If $\sigma, \tau \in K$, then $\sigma \cap \tau$ is either empty or a face of both $\sigma$ and $\tau$. [/definition] [definition: Vertices of a Complex] The **vertices** of a simplicial complex $K$ are its $0$-simplices, collected in the set $V_K$. [/definition] Condition 1 says the complex is closed under taking faces; condition 2 says distinct simplices only meet along shared faces. Together, these prevent the pathological gluings that would arise from, say, two triangles meeting at a single interior point or along a partial edge. [example: Valid and Invalid Simplicial Complexes] A collection consisting of a tetrahedron sharing a face with another triangle, with a separate triangle connected by a single edge, with a further isolated edge, constitutes a valid simplicial complex. Every intersection of any two simplices in the collection is either empty or a face of both. For an invalid example: take two triangles whose boundaries overlap in a single interior point (not a vertex). Their intersection is not a face of either triangle, violating condition 2. Similarly, two triangles meeting along a complete edge where one triangle includes the edge as a face but the other has it only as a subset of a longer edge would fail condition 2. [/example] [illustration:simplicial-complex-valid-invalid] ### Polyhedra and Dimension A simplicial complex $K$ is an abstract combinatorial object — it is a *collection* of simplices, not itself a subspace of $\mathbb{R}^m$. The associated subspace is given a separate name. [definition: Polyhedron] The **polyhedron** of $K$, denoted $|K|$, is the union of all simplices in $K$ as a subspace of $\mathbb{R}^m$. [/definition] The distinction matters because two different simplicial complexes can give the same polyhedron: for example, subdividing a line segment by adding a midpoint produces a complex with three vertices and two edges, but the polyhedron is still the same interval. Since each simplex is compact and the complex is finite, the polyhedron $|K|$ is always compact and Hausdorff. [definition: Dimension and Skeleton] The **dimension** of a simplicial complex $K$ is the maximum dimension among all simplices in $K$. The **$d$-skeleton** $K^{(d)}$ is the sub-complex consisting of all simplices of dimension at most $d$. [/definition] The $d$-skeleton captures the "skeleton" of the complex at each level. The $0$-skeleton is just the set of vertices; the $1$-skeleton is the graph formed by vertices and edges; and so on. ### Triangulations Many spaces we care about — spheres, projective spaces, surfaces — are not defined as simplicial complexes. We can bring them under the simplicial framework by *triangulating* them. [definition: Triangulation] A **triangulation** of a topological space $X$ is a homeomorphism $h: |K| \to X$, where $K$ is some simplicial complex. [/definition] [example: The Simplicial $(n-1)$-Sphere] Let $\sigma$ be the standard $n$-simplex. Its boundary $\partial \sigma$ is homeomorphic to $S^{n-1}$. When $n = 3$, the boundary of the solid tetrahedron is a union of four triangular faces glued along edges; this is homeomorphic to $S^2$. This is called the **simplicial $(n-1)$-sphere**. To see the homeomorphism more concretely in the case $n = 2$: the standard $2$-simplex has vertices $e_0, e_1, e_2$ in $\mathbb{R}^3$. Its boundary consists of three edges $\langle e_0, e_1 \rangle$, $\langle e_1, e_2 \rangle$, and $\langle e_0, e_2 \rangle$, forming a triangle. A triangle is topologically a circle $S^1$. The homeomorphism is the map that sends each boundary point to the corresponding point on $S^1$ via radial projection from the centroid. [/example] [example: The Cross-Polytope Triangulation of $S^n$] In $\mathbb{R}^{n+1}$, consider all $n$-simplices of the form $\langle \pm e_0, \pm e_1, \ldots, \pm e_n \rangle$ ranging over all $2^{n+1}$ sign combinations. Their union forms a simplicial complex $K$ with $|K| \cong S^n$. For $n = 2$, this gives eight triangles — the faces of a regular octahedron — whose union is the $2$-sphere. This triangulation has a particularly attractive property: it is invariant under the antipodal map $x \mapsto -x$. Therefore it descends to a triangulation of $\mathbb{R}P^n$, where vertices are equivalence classes of the $2(n+1)$ vertices under the antipodal identification. [/example] ### Simplicial Maps Having defined the objects, we turn to the maps between them. The central principle of algebraic topology is always to study not just spaces in isolation, but maps between them and the functorial structure they induce. [definition: Simplicial Map] A **simplicial map** $f: K \to L$ is a function $f: V_K \to V_L$ on vertex sets such that for every simplex $\langle a_0, \ldots, a_n \rangle \in K$, the image set $\{f(a_0), \ldots, f(a_n)\}$ spans a simplex of $L$. [/definition] The decisive advantage of simplicial maps is that they are entirely determined by finitely many data: the images of the finitely many vertices. There is no need to verify continuity or to write down a formula for intermediate points. Note carefully that the image $\{f(a_0), \ldots, f(a_n)\}$ is a *set*, not a tuple: repetitions are allowed. A simplex can be "collapsed" by a simplicial map to a lower-dimensional simplex when multiple vertices are sent to the same vertex. [example: Simplicial Maps on the Standard $2$-Simplex] Let $K$ be the standard $2$-simplex with vertices $a_0, a_1, a_2$, and let $L$ consist of a single edge with vertices $b_0, b_1$. The assignment $f(a_0) = f(a_1) = b_0$, $f(a_2) = b_1$ is a valid simplicial map: the face $\langle a_0, a_1 \rangle$ maps to $\{b_0, b_0\} = \{b_0\}$, which spans the $0$-simplex $b_0$ in $L$; the face $\langle a_1, a_2 \rangle$ maps to $\{b_0, b_1\}$, which spans the edge; the face $\langle a_0, a_2 \rangle$ similarly spans the edge. The full $2$-simplex $\{a_0, a_1, a_2\}$ maps to $\{b_0, b_1\}$, which spans the $1$-simplex (not a $2$-simplex, because the set has only two elements). So the triangle is collapsed onto the edge. However, the assignment $f(a_0) = b_0$, $f(a_1) = b_1$, $f(a_2) = b_0$ with $L$ having only the vertex $b_0$ and the vertex $b_1$ but *no* edge between them would fail: $\langle a_0, a_1 \rangle$ maps to $\{b_0, b_1\}$, which must span a simplex in $L$, but there is no edge $\langle b_0, b_1 \rangle$ in $L$. [/example] Simplicial maps induce continuous maps of polyhedra by linear extension. This is the content of the following lemma, which also shows that the construction is functorial. [quotetheorem:1913] [citeproof:1913] This theorem means simplicial complexes and simplicial maps form a category, and geometric realization is a functor to topological spaces and continuous maps. One important structural fact, used later in the proof of the simplicial approximation theorem, is that every point in a polyhedron has a unique interior simplex. [quotetheorem:1914] The uniqueness here is not entirely obvious: the boundary of a simplex is shared by multiple simplices, but the interior is not shared. This makes the interior simplex a well-defined notion and allows barycentric coordinates to be read off unambiguously. ## Simplicial Approximation Simplicial maps are tractable, but the spaces we care about admit many continuous maps that are not simplicial with respect to any fixed triangulation. The question is: given a continuous map $f: |K| \to |L|$, can we find a simplicial map that is, in some sense, close to $f$? The answer is yes — after subdividing $K$ sufficiently finely. The key notion is that of an "approximation" which records, for each vertex, where it must be sent so that the homotopy class is preserved. ### Open Stars and the Approximation Condition [definition: Open Star and Link] Let $K$ be a simplicial complex and $v \in V_K$ a vertex. The **open star** of $v$ is \begin{align*} \operatorname{St}_K(v) = \bigcup_{v \in \sigma \in K} \mathring{\sigma}, \end{align*} the union of the interiors of all simplices containing $v$. The **link** of $v$, written $\operatorname{Lk}_K(v)$, is the union of all simplices that do not contain $v$ but are faces of a simplex that does contain $v$. [/definition] The open star of $v$ is an open neighborhood of $v$ in $|K|$: it consists of all points lying in some simplex that has $v$ as a vertex, minus the faces that do not contain $v$. Intuitively, it is the "star-shaped" region around $v$. The open stars $\{\operatorname{St}_K(v)\}_{v \in V_K}$ cover all of $|K|$, since every point lies in the interior of some simplex whose vertices each have stars containing that point. [definition: Simplicial Approximation] Let $f: |K| \to |L|$ be a continuous map. A function $g: V_K \to V_L$ is a **simplicial approximation** to $f$ if for every vertex $v \in V_K$: \begin{align*} f(\operatorname{St}_K(v)) \subseteq \operatorname{St}_L(g(v)). \end{align*} [/definition] This condition says that $g$ and $f$ "agree" on the open star level: wherever $f$ sends the star of $v$, the star of $g(v)$ covers the image. The following theorem shows this is the right definition — the approximation condition forces $g$ to be a simplicial map, and the resulting map $|g|$ is homotopic to $f$. [quotetheorem:1915] [citeproof:1915] This result is the key conceptual payoff: up to homotopy, every continuous map is as good as a simplicial one — provided we can find an approximation. The obstacle is that a given triangulation may not be fine enough. The solution is barycentric subdivision. ### Barycentric Subdivision The idea is to refine a simplicial complex by replacing each simplex with a collection of smaller simplices, each of which fits inside the original. We do this by introducing the barycenter of each simplex as a new vertex. [definition: Barycenter] The **barycenter** of a simplex $\sigma = \langle a_0, \ldots, a_n \rangle$ is \begin{align*} \hat{\sigma} = \frac{1}{n+1}\sum_{i=0}^n a_i, \end{align*} the average of the vertices. It is the unique point in $\mathring{\sigma}$ with all barycentric coordinates equal to $\frac{1}{n+1}$. [/definition] [definition: Barycentric Subdivision] The **first barycentric subdivision** $K'$ of $K$ is the simplicial complex \begin{align*} K' = \left\{ \langle \hat{\sigma}_0, \hat{\sigma}_1, \ldots, \hat{\sigma}_k \rangle \;\Big|\; \sigma_0 < \sigma_1 < \cdots < \sigma_k,\; \sigma_i \in K \right\}, \end{align*} where $\sigma_i < \sigma_j$ means $\sigma_i$ is a proper face of $\sigma_j$. The $r$th barycentric subdivision $K^{(r)}$ is defined inductively: $K^{(0)} = K$ and $K^{(r)} = (K^{(r-1)})'$. [/definition] The vertices of $K'$ are the barycenters $\hat{\sigma}$ of all simplices $\sigma \in K$ (including vertices, which are their own barycenters). A new simplex in $K'$ is determined by a strictly increasing chain of faces $\sigma_0 \subsetneq \sigma_1 \subsetneq \cdots \subsetneq \sigma_k$ in $K$; the corresponding simplex of $K'$ has these barycenters as vertices. [illustration:barycentric-subdivision-triangle] [quotetheorem:1917] The proof that $K'$ satisfies the two axioms of a simplicial complex is a verification of compatibility conditions for chains of faces; the course omits the details. The key consequence is that $K'$ provides a genuinely finer decomposition of the same underlying space. The identity map $|K'| \to |K|$ is *not* a simplicial map (the vertices of $K'$ are barycenters, which are not vertices of $K$ in general). However, for each simplex $\sigma \in K$, we can choose any one of its vertices $v_\sigma \in \sigma$ and define $g: K' \to K$ by $g(\hat{\sigma}) = v_\sigma$. This assignment is a simplicial map and serves as a simplicial approximation to the identity. This observation will be essential in the proof of homotopy invariance of homology. To make the approximation theorem work, we need to measure how fine a subdivision is. [definition: Mesh] The **mesh** of a simplicial complex $K$ is \begin{align*} \mu(K) = \max\left\{ \|v_0 - v_1\| \;:\; \langle v_0, v_1 \rangle \in K \right\}, \end{align*} the maximum length of any edge in $K$. [/definition] The mesh controls the diameter of the open stars: each open star $\operatorname{St}_K(v)$ is contained in a ball of radius $\mu(K)$ around $v$. The following lemma shows that repeated barycentric subdivision drives the mesh to zero. [quotetheorem:1918] The proof is a geometric estimate on how much smaller the edges of the subdivision become compared to those of $K$; it is a direct calculation which the course omits. The bound $\frac{n}{n+1} < 1$ is the essential quantitative input. Since $\frac{n}{n+1} < 1$, the ratio is a genuine contraction factor. As $r$ increases, the mesh decreases geometrically, providing arbitrarily fine subdivisions of any simplicial complex. ### The Simplicial Approximation Theorem We now have all the ingredients for the main theorem of the chapter. [quotetheorem:1919] [citeproof:1919] The simplicial approximation theorem is the technical backbone that justifies the entire simplicial homology programme. The theorem guarantees that up to homotopy — and homotopy is exactly the equivalence relation that homology respects — every continuous map between polyhedra can be represented by a simplicial map after sufficiently many subdivisions. This means homology, once defined on simplicial complexes and simplicial maps, automatically extends to all continuous maps between spaces that admit triangulations. [remark: What the Theorem Achieves] The theorem has a subtle but important structure: we do not subdivide the *target* $L$, only the domain $K$. This is essential for the construction to be functorial: the map goes from a refinement of $K$ to the *same* complex $L$. Since barycentric subdivision does not change the underlying polyhedron ($|K^{(r)}| = |K|$), the induced continuous map $|g|: |K| \to |L|$ is homotopic to the original $f: |K| \to |L|$. So, at the level of homotopy classes of maps — the level relevant to homology — every continuous map is equivalent to a simplicial one. [/remark] The framework built in this chapter — simplices, complexes, simplicial maps, barycentric subdivision, and the approximation theorem — provides the precise combinatorial language in which simplicial homology is defined. In the next chapter, we will use the face structure of simplicial complexes to define chain groups and boundary operators, culminating in the homology groups $H_n(K)$. # 8. Simplicial Homology The fundamental group captures a great deal of topological information — it detects loops that cannot be contracted, distinguishes surfaces of different genus, and provides the algebraic backbone for covering space theory. Yet it has a significant limitation: it is inherently one-dimensional, encoding only the behaviour of paths up to homotopy. A sphere of dimension two or higher has no interesting fundamental group, yet it is not contractible, and any reasonable theory of holes should see its hollow interior. Simplicial homology remedies this by producing a whole sequence of abelian groups $H_0(K), H_1(K), H_2(K), \ldots$ associated to a simplicial complex $K$, where $H_n(K)$ measures $n$-dimensional holes. The price of working in all dimensions simultaneously is that the definitions appear more algebraic and less geometric than $\pi_1$, but the trade-off is a theory that is vastly more computable: every step reduces to linear algebra over the integers. ## Simplicial Homology The central problem this section addresses is the following: how can we assign a meaningful algebraic invariant to a simplicial complex that detects not just connected components and one-dimensional loops, but higher-dimensional voids? The answer is to assign to each dimension $n$ a chain group formed by formal sums of $n$-simplices, equip these groups with boundary maps, and then measure how many cycles fail to be boundaries. Throughout this chapter $K$ denotes a simplicial complex and $V_K$ its vertex set. We work primarily with abelian groups, which should be thought of as $\mathbb{Z}$-modules; the theory works over any commutative ring, and we will occasionally note which statements simplify over $\mathbb{Q}$. When we defined simplices, we identified two orderings of vertices as the same simplex whenever they spanned the same geometric object. For homology, we need to remember which direction we traverse a simplex — the algebraic boundary map requires a consistent orientation. [definition: Oriented Simplex] An **oriented $n$-simplex** in a simplicial complex $K$ is an $(n+1)$-tuple $(a_0, \ldots, a_n)$ of vertices $a_i \in V_K$ such that $\langle a_0, \ldots, a_n \rangle \in K$. Two tuples $(a_0, \ldots, a_n)$ and $(a_{\pi(0)}, \ldots, a_{\pi(n)})$ represent the **same** oriented simplex when $\pi \in S_n$ is an even permutation. We write $\sigma$ for an oriented simplex and $\bar{\sigma}$ for the same simplex with the opposite orientation. [/definition] The partition of permutations into even and odd is exactly right here: there are two equivalence classes of orderings, corresponding to the two possible orientations. In dimension one, the two orientations of an edge $(a_0, a_1)$ are simply the two directions along it. In dimension two, the two orientations of a triangle $(v_0, v_1, v_2)$ correspond to the two circular orderings — clockwise and counterclockwise — of the boundary edges. [example: Orientations of a 2-Simplex] As oriented $2$-simplices, $(v_0, v_1, v_2)$ and $(v_1, v_2, v_0)$ are equal (their difference is the even permutation $(012) \mapsto (120)$, a cyclic shift), but they differ from $(v_2, v_1, v_0)$, whose vertex ordering is related by an odd transposition. The two distinct orientations of a triangle can be visualised as counterclockwise and clockwise cyclic orderings of the vertices. [/example] [illustration:oriented-2-simplex] With oriented simplices in hand, we can form the groups that carry all of the homological information. [definition: Chain Group] Let $K$ be a simplicial complex and $n \geq 0$. Choose an orientation on each $n$-simplex of $K$ — that is, for each $n$-simplex, fix an ordering of its vertices up to even permutation. The **$n$th chain group** $C_n(K)$ is the free abelian group with basis the set $\{\sigma_1, \ldots, \sigma_\ell\}$ of oriented $n$-simplices of $K$. Concretely, $C_n(K) \cong \mathbb{Z}^\ell$ and a general element is a formal linear combination $\sum_{i} m_i \sigma_i$ with $m_i \in \mathbb{Z}$. For each oriented simplex $\sigma$ with $n \geq 1$, we identify $-\sigma$ with the opposite orientation $\bar{\sigma}$. We set $C_n(K) = 0$ for $n < 0$ by convention. [/definition] The choice of orientation for each simplex is arbitrary but must be made; different choices produce isomorphic chain groups via the map $\sigma_i \mapsto \pm \sigma_i$, so the homology groups we define below are independent of the choice. [example: Chains as Paths] Elements of $C_1(K)$ are naturally interpreted as formal combinations of oriented edges — that is, as generalised paths. If $K$ contains vertices $v_0, v_1, v_2$ and edges $(v_0, v_1), (v_1, v_2), (v_2, v_0)$, then the combination $c = (v_0, v_1) + (v_1, v_2) + (v_2, v_0)$ represents the loop traversing all three edges once in the consistent direction. One can also form combinations like $2\sigma_1 - \sigma_2 + \sigma_3$ that have no direct geometric path interpretation, but these formal sums are precisely what allows us to capture all homological information algebraically. [/example] The crucial piece of structure is the boundary map, which extracts the $(n-1)$-dimensional boundary of an $n$-simplex. [definition: Boundary Homomorphism] For each $n \geq 1$, the **$n$th boundary homomorphism** \begin{align*} d_n : C_n(K) &\to C_{n-1}(K) \end{align*} is defined on oriented simplices by \begin{align*} d_n(a_0, \ldots, a_n) &= \sum_{i=0}^{n} (-1)^i (a_0, \ldots, \hat{a}_i, \ldots, a_n), \end{align*} where $(a_0, \ldots, \hat{a}_i, \ldots, a_n)$ denotes the $(n-1)$-simplex obtained by omitting the vertex $a_i$. Extend $d_n$ linearly to all of $C_n(K)$. Set $d_0 : C_0(K) \to C_{-1}(K) = 0$ to be the zero map. [/definition] The alternating signs are essential: they ensure that adjacent faces cancel correctly. For an edge $(v_0, v_1)$, the boundary is $v_1 - v_0$ — the terminal vertex minus the initial vertex, consistent with signed boundary. For an oriented triangle $(v_0, v_1, v_2)$, the boundary is the signed sum of edges \begin{align*} d_2(v_0, v_1, v_2) &= (v_1, v_2) - (v_0, v_2) + (v_0, v_1), \end{align*} which equals $(v_0, v_1) + (v_1, v_2) + (v_2, v_0)$: the boundary traversed consistently. [illustration:boundary-of-2-simplex] The fundamental algebraic property of the boundary maps is the following: [quotetheorem:1920] [citeproof:1920] The identity $d_{n-1} \circ d_n = 0$ is exactly the condition $\operatorname{im} d_{n+1} \subseteq \ker d_n$, which makes the following definition well-posed. [definition: Simplicial Homology Group] The **$n$th simplicial homology group** of $K$ is \begin{align*} H_n(K) &= \frac{\ker d_n}{\operatorname{im} d_{n+1}}. \end{align*} [/definition] The groups $\ker d_n$ and $\operatorname{im} d_{n+1}$ have names that are worth fixing. [definition: Chains, Cycles, and Boundaries] Elements of $C_n(K)$ are called **$n$-chains**. Elements of $Z_n(K) := \ker d_n$ are called **$n$-cycles**. Elements of $B_n(K) := \operatorname{im} d_{n+1}$ are called **$n$-boundaries**. [/definition] The geometric picture is as follows. A cycle $c \in Z_n(K)$ is an $n$-chain with no boundary — something that closes up on itself, like a loop in one dimension or a closed surface in two. A boundary $c \in B_n(K)$ is a cycle that happens to be the boundary of some $(n+1)$-chain, meaning it bounds a higher-dimensional region. Two cycles represent the same homology class precisely when they differ by a boundary, i.e., when one can be deformed into the other by filling in an $(n+1)$-dimensional region. Thus $H_n(K)$ counts the $n$-dimensional holes: the cycles that have not been filled in. [example: Homology of the Standard 1-Sphere] Let $K$ be the simplicial complex consisting of vertices $e_0, e_1, e_2$ and edges $\langle e_0, e_1 \rangle, \langle e_1, e_2 \rangle, \langle e_2, e_0 \rangle$ — a triangulated circle. Choose orientations $(e_0, e_1), (e_1, e_2), (e_2, e_0)$. The chain groups are \begin{align*} C_0(K) &\cong \mathbb{Z}^3, & C_1(K) &\cong \mathbb{Z}^3, \end{align*} and all others are $0$. The boundary map $d_1 : C_1(K) \to C_0(K)$ is represented in the chosen bases by the matrix \begin{align*} \begin{pmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{pmatrix}. \end{align*} The image of $d_1$ is the $\mathbb{Z}$-span of the columns, which has rank $2$ (the third column is the negative of the sum of the first two). Therefore $H_0(K) = \mathbb{Z}^3 / \operatorname{im} d_1 \cong \mathbb{Z}$. For $H_1(K)$: since there are no $2$-simplices, $\operatorname{im} d_2 = 0$, so $H_1(K) = \ker d_1$. The kernel of $d_1$ is the $\mathbb{Z}$-span of $(e_0, e_1) + (e_1, e_2) + (e_2, e_0)$, since one checks that $d_1$ applied to this combination gives $(e_1 - e_0) + (e_2 - e_1) + (e_0 - e_2) = 0$. Hence $H_1(K) \cong \mathbb{Z}$. The generator of $H_1(K)$ is the fundamental loop of the circle. The fact that $H_1(K) \neq 0$ reflects the presence of the one-dimensional hole in the interior of the triangle. [/example] [example: Homology of the Standard 2-Simplex] Let $L$ be the simplicial complex consisting of the triangle $(e_0, e_1, e_2)$ and all its faces — a filled triangle. The chain groups are \begin{align*} C_0(L) &\cong \mathbb{Z}^3, & C_1(L) &\cong \mathbb{Z}^3, & C_2(L) &\cong \mathbb{Z}, \end{align*} with $C_k(L) = 0$ for $k \geq 3$. The map $d_1$ is the same matrix as in the previous example. The new boundary map is $d_2 : C_2(L) \to C_1(L)$, which sends the generator $(e_0, e_1, e_2)$ to \begin{align*} d_2(e_0, e_1, e_2) &= (e_1, e_2) - (e_0, e_2) + (e_0, e_1) = (e_0, e_1) + (e_1, e_2) + (e_2, e_0). \end{align*} Since $d_0 = 0$, we have $H_0(L) = C_0(L)/\operatorname{im} d_1 \cong \mathbb{Z}$, just as before — $L$ is path-connected. For $H_1(L)$: the kernel of $d_1$ is spanned by $(e_0, e_1) + (e_1, e_2) + (e_2, e_0)$, and the image of $d_2$ is also spanned by precisely this element. Therefore \begin{align*} H_1(L) &= \ker d_1 / \operatorname{im} d_2 \cong \mathbb{Z}/\mathbb{Z} = 0. \end{align*} The hole has been filled: the bounding loop in $K$ is now the boundary of the $2$-chain $L$. Finally, $H_2(L) = \ker d_2 / \operatorname{im} d_3 = \ker d_2$. Since $d_2$ sends a non-zero generator to a non-zero element, $d_2$ is injective, so $\ker d_2 = 0$ and $H_2(L) = 0$. [/example] These two calculations illustrate the key point: filling in the triangle kills the $H_1$ class and introduces no new $H_2$ class, because there is no enclosed two-dimensional void. We now have enough geometric intuition to proceed to the algebraic machinery needed for systematic computation. A remark on $H_0$ is worth recording before moving on. [quotetheorem:1921] [citeproof:1921] This exactly mirrors the role of $\pi_0$ for the fundamental group. For path-connected spaces we always have $H_0 \cong \mathbb{Z}$, so in practice we will often focus on the higher homology groups. ## Homological Algebra Having seen the definition in action, we now build the algebraic formalism that makes systematic computation possible. The key idea is to axiomatize the structure present in $(C_n(K), d_n)$: a sequence of groups with maps that square to zero. [definition: Chain Complex] A **chain complex** $C_\bullet$ is a sequence of abelian groups $\ldots, C_2, C_1, C_0$ equipped with homomorphisms $d_n : C_n \to C_{n-1}$ satisfying $d_{n-1} \circ d_n = 0$ for all $n$. The maps $d_n$ are called the **differentials** of $C_\bullet$. We represent this as a diagram: \begin{align*} \cdots \longrightarrow C_2 \xrightarrow{d_2} C_1 \xrightarrow{d_1} C_0 \xrightarrow{d_0} 0. \end{align*} [/definition] The condition $d^2 = 0$ is exactly the condition required to form homology groups: the image of each differential lands inside the kernel of the next. [definition: Homology of a Chain Complex] For a chain complex $C_\bullet$, define: - The group of **$n$-cycles**: $Z_n(C) := \ker d_n \subseteq C_n$. - The group of **$n$-boundaries**: $B_n(C) := \operatorname{im} d_{n+1} \subseteq C_n$. The **$n$th homology group** is \begin{align*} H_n(C) &:= Z_n(C)/B_n(C). \end{align*} [/definition] Since $B_n(C) \subseteq Z_n(C)$ by $d^2 = 0$, this quotient is well-defined. The simplicial homology $H_n(K)$ is then the homology of the chain complex $(C_n(K), d_n)$. Maps between chain complexes should respect the differential structure. This is the algebraic analogue of a continuous map respecting the topology. [definition: Chain Map] A **chain map** $f_\bullet : C_\bullet \to D_\bullet$ is a collection of homomorphisms $f_n : C_n \to D_n$ such that the following diagram commutes for all $n$: \begin{align*} d_n^D \circ f_n &= f_{n-1} \circ d_n^C. \end{align*} In other words, the maps $f_n$ interleave with the differentials: $f$ commutes with $d$. [/definition] Chain maps are the morphisms in the category of chain complexes. Their key property is that they descend to maps on homology. [quotetheorem:1922] [citeproof:1922] This result says that the assignment $C_\bullet \mapsto H_n(C_\bullet)$ is functorial: it sends chain complexes to abelian groups and chain maps to group homomorphisms, respecting composition and identities. The notation $f_*$ for the induced map on homology is standard. We now introduce the algebraic counterpart of a homotopy between continuous maps. [definition: Chain Homotopy] A **chain homotopy** between chain maps $f_\bullet, g_\bullet : C_\bullet \to D_\bullet$ is a collection of homomorphisms $h_n : C_n \to D_{n+1}$ satisfying \begin{align*} g_n - f_n &= d_{n+1}^D \circ h_n + h_{n-1} \circ d_n^C \end{align*} for all $n$. We write $f_\bullet \simeq g_\bullet$ when such an $h_\bullet$ exists. [/definition] The geometric motivation for this definition comes from actual homotopies of simplicial maps. If $H : |K| \times [0,1] \to |L|$ is a homotopy from a simplicial map $f$ to a simplicial map $g$, one can define $h_n(\sigma) = H(\sigma \times [0,1])$, interpreted as an $(n+1)$-chain. Computing its boundary yields exactly the chain homotopy relation after accounting for the orientations of the top and bottom faces and the vertical sides. The definition is thus the algebraic shadow of a geometric homotopy. [quotetheorem:1923] [citeproof:1923] Chain homotopy equivalence is the right notion of isomorphism for chain complexes — the algebraic analogue of homotopy equivalence for topological spaces. [definition: Chain Homotopy Equivalence] Chain complexes $C_\bullet$ and $D_\bullet$ are **chain homotopy equivalent** if there exist chain maps $f_\bullet : C_\bullet \to D_\bullet$ and $g_\bullet : D_\bullet \to C_\bullet$ such that \begin{align*} f_\bullet \circ g_\bullet \simeq \operatorname{id}_{D_\bullet}, \qquad g_\bullet \circ f_\bullet \simeq \operatorname{id}_{C_\bullet}. \end{align*} [/definition] [quotetheorem:1925] [citeproof:1925] This theorem is what makes chain homotopy equivalences so useful in practice: to compute the homology of a complicated complex, it suffices to find a chain homotopy equivalent complex whose homology is known. We will exploit this immediately in the next section. Some routine but important formal properties of chain homotopies and induced maps are worth recording. [remark: Functoriality of $H_n$] The following hold and can be verified from the definitions. Chain homotopy equivalence is an equivalence relation on chain maps. If $a_\bullet : A_\bullet \to C_\bullet$ is a chain map and $f_\bullet \simeq g_\bullet : C_\bullet \to D_\bullet$, then $f_\bullet \circ a_\bullet \simeq g_\bullet \circ a_\bullet$. For composable chain maps, $(g_\bullet \circ f_\bullet)_* = g_* \circ f_*$, and $(\operatorname{id}_{C_\bullet})_* = \operatorname{id}_{H_*(C)}$. The last two properties together say that $H_n$ is a functor from chain complexes to abelian groups. [/remark] ## Homology Calculations The abstract machinery now pays off in concrete computations. The key results are the homology of a cone (which collapses to a point) and the homology of the boundary of a simplex (which sees a top-dimensional hole). The first step is to note that a simplicial map induces a chain map, hence a map on homology. [quotetheorem:1926] [citeproof:1926] [definition: Cone] A simplicial complex $K$ is a **cone** with **cone point** $v_0 \in V_K$ if every simplex of $K$ is either a simplex of $\operatorname{Lk}_K(v_0)$, or has $v_0$ as a vertex. [/definition] Geometrically, a cone is a space that can be contracted to its apex by pulling everything toward $v_0$. The following lemma makes this algebraically precise. [illustration:cone-complex] [quotetheorem:1927] [citeproof:1927] The homology of a cone is entirely concentrated in degree zero — there are no higher-dimensional holes because one can always contract toward the apex. This immediately gives us the homology of the standard simplex and its boundary. [quotetheorem:1928] [citeproof:1928] [quotetheorem:1929] [citeproof:1929] This result is one of the most important computations in the course. It shows that $S^{n-1}$ has a non-trivial homology class in degree $n-1$, generated by its fundamental cycle — the entire boundary sphere. The fundamental group could not see spheres of dimension two or higher, but $H_{n-1}$ detects the void enclosed by $S^{n-1}$ for all $n \geq 2$. This will eventually allow us to distinguish $\mathbb{R}^n$ from $\mathbb{R}^m$ when $n \neq m$, a result that the fundamental group alone cannot achieve. ## The Mayer-Vietoris Sequence The Mayer-Vietoris theorem is the homological counterpart of the Seifert–van Kampen theorem: it computes the homology of a union from the homology of the pieces and their intersection. Before stating it, we need to introduce exact sequences — the algebraic notion that makes the output of the theorem precise. [definition: Exact Sequence] A pair of homomorphisms \begin{align*} A \xrightarrow{f} B \xrightarrow{g} C \end{align*} is **exact at $B$** if $\operatorname{im} f = \ker g$. A longer sequence \begin{align*} \cdots \to A_{i} \xrightarrow{f_i} A_{i+1} \xrightarrow{f_{i+1}} A_{i+2} \to \cdots \end{align*} is **exact** if it is exact at every $A_i$, i.e., $\ker f_i = \operatorname{im} f_{i-1}$ for all $i$. [/definition] Exactness is a strengthening of the condition $d^2 = 0$ in a chain complex: instead of requiring only $\operatorname{im} d \subseteq \ker d$, we require equality. Homology groups therefore measure the failure of a sequence to be exact: $H_n = 0$ if and only if the chain complex is exact at degree $n$. [definition: Short Exact Sequence] A **short exact sequence** is an exact sequence of the form \begin{align*} 0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0. \end{align*} [/definition] Unpacking exactness at each term: the map $f$ is injective (its kernel equals the image of $0 \to A$, which is $\{0\}$); the map $g$ is surjective (its image equals the kernel of $C \to 0$, which is all of $C$); and $\operatorname{im} f = \ker g$ says that $C \cong B/\operatorname{im} f$. A short exact sequence encodes the data of a subgroup $A \hookrightarrow B$ and its quotient $B/A \cong C$. [definition: Short Exact Sequence of Chain Complexes] A **short exact sequence of chain complexes** consists of chain maps $i_\bullet, j_\bullet$ such that \begin{align*} 0 \to A_\bullet \xrightarrow{i_\bullet} B_\bullet \xrightarrow{j_\bullet} C_\bullet \to 0 \end{align*} and for each $k$, the sequence $0 \to A_k \xrightarrow{i_k} B_k \xrightarrow{j_k} C_k \to 0$ is exact. [/definition] A short exact sequence of chain complexes does not produce short exact sequences in homology — instead, it produces a long exact sequence via a remarkable connecting homomorphism. This is the content of the Snake Lemma. [quotetheorem:1930] [citeproof:1930] The proof technique — tracing elements through a commutative diagram to define new ones and verify properties — is called **diagram chasing** and appears throughout homological algebra. The mental image is of an element moving through the grid of the diagram, changing groups at each step. The strategy for applications is that knowing any four consecutive terms in the long exact sequence pins down the fifth, at least up to extension. With the Snake Lemma in place, the Mayer-Vietoris theorem follows almost immediately by writing down the right short exact sequence of chain complexes. [quotetheorem:1931] [citeproof:1931] Notice that Mayer-Vietoris, unlike Seifert–van Kampen, does not require the intersection $L = M \cap N$ to be path-connected. The fundamental group has difficulty handling non-connected basepoints, but homology groups have no such restriction — they are global invariants, not based at any particular point. [remark: Using the Mayer-Vietoris Sequence] The practical power of Mayer-Vietoris is inductive: if we can decompose $K$ into pieces $M$ and $N$ whose homology groups we already know, and we know the homology of the intersection, then the long exact sequence — combined with exactness — pins down the homology of $K$. Often, we know all terms except the one we want; exactness then forces the unknown term. The connecting homomorphism $\partial_*$ must be understood from the proof of the Snake Lemma to carry out these arguments correctly. [/remark] ## Continuous Maps and Homotopy Invariance The homology groups $H_n(K)$ have been defined using the simplicial structure of $K$, and the maps we have induced so far come only from simplicial maps. The central goal of this section is to lift both of these restrictions: we want $H_n(K)$ to depend only on the underlying space $|K|$, not on the choice of triangulation, and we want continuous maps — not just simplicial ones — to induce well-defined maps on homology. Achieving this requires careful work with simplicial approximation and barycentric subdivision. The key intermediate notion connects simplicial maps to homotopies. [definition: Contiguous Maps] Simplicial maps $f, g : K \to L$ are **contiguous** if for each simplex $\sigma \in K$, the simplices $f(\sigma)$ and $g(\sigma)$ are both faces of some common simplex $\tau \in L$. [/definition] Contiguity is a combinatorial version of being "close": the images of any simplex under $f$ and $g$ fit together inside a single simplex of $L$. This is precisely the condition needed for the two maps to be chain-homotopic. [quotetheorem:1933] [citeproof:1933] The next result connects the abstract notion of contiguity to the geometric situation of simplicial approximation. [quotetheorem:1934] [citeproof:1934] To extend from simplicial to continuous maps, we must also show that barycentric subdivision does not change the homology. The argument uses the following fact about approximations to the identity. [quotetheorem:1935] The map $a$ is highly non-unique — there are many choices of vertex for each simplex — but all such choices are contiguous to each other, hence give the same induced map on homology. [quotetheorem:1936] [citeproof:1936] [quotetheorem:1938] The Five Lemma follows from diagram chasing and is a standard tool of homological algebra; it appears as an exercise on the example sheet. We can now define the induced map for a continuous map between polyhedra. [quotetheorem:1939] The proof uses the contiguity results above — different choices of approximation give contiguous maps, hence the same map on homology — and the fact that subdivision isomorphisms are natural. The details are omitted here. [quotetheorem:1942] [citeproof:1942] This result establishes that $H_n(K)$ is an invariant of the underlying topological space $|K|$, not of the simplicial structure chosen. For instance, the calculation $H_{n-1}(\partial \Delta^n) \cong \mathbb{Z}$ for the boundary of the standard $n$-simplex gives the same answer for any triangulation of $S^{n-1}$. To pass from homeomorphism to homotopy invariance, we use a two-step strategy: first show that maps which are $\varepsilon$-close induce the same map on homology, then decompose any homotopy into finitely many small steps. [quotetheorem:1943] [citeproof:1943] [quotetheorem:1944] [citeproof:1944] This is the culminating result of the technical development. Homology groups are homotopy invariants. [definition: $h$-Triangulation and Homology of a Space] An **$h$-triangulation** of a topological space $X$ is a simplicial complex $K$ together with a homotopy equivalence $h : |K| \to X$. We define $H_n(X) := H_n(K)$ for all $n$. [/definition] The previous theorem guarantees that this definition does not depend on the choice of $K$ or $h$: any two $h$-triangulations of $X$ give isomorphic homology groups, since the transition maps are homotopy equivalences and hence isomorphisms on homology. This extends the reach of simplicial homology well beyond compact simplicial complexes — for example, open balls and other non-compact spaces that are homotopy equivalent to a simplicial complex now have well-defined homology groups. The machinery developed here — chain complexes, chain maps, chain homotopies, the Snake Lemma, and Mayer-Vietoris — forms the backbone of modern algebraic topology. In the second half of this chapter we will apply it to compute the homology of surfaces, use it to prove classical theorems such as Brouwer's fixed point theorem, and explore the relationship between homology and the fundamental group. ## Homology of Spheres and Applications The sphere $S^{n-1}$ is one of the most important spaces in topology, and its homology groups are the engine behind several of the deepest classical results in the subject. Once we have computed these groups, we can translate topological questions — does this map have a fixed point? does this vector field vanish? — into purely algebraic ones that homology answers decisively. [quotetheorem:1945] [citeproof:1945] This single computation has immediate and far-reaching consequences. The key point is that $H_{n-1}(D^n) = 0$ while $H_{n-1}(S^{n-1}) \cong \mathbb{Z}$, and this algebraic gap is what forces fixed points to exist and vector fields to vanish. [quotetheorem:1947] The proof, which appears on Example Sheet 4, removes a point from each space and compares the homology of the resulting punctured spaces: $\mathbb{R}^n \setminus \{0\} \simeq S^{n-1}$, whose $(n-1)$st homology group distinguishes dimensions. This is a case where homology cleanly resolves a question that would be extraordinarily difficult to settle by other means. ### Brouwer's Fixed Point Theorem in All Dimensions The two-dimensional Brouwer fixed point theorem was proved earlier using the fundamental group. Homology now gives the same result in every dimension, by exactly the same style of argument — but with $H_{n-1}$ replacing $\pi_1$. [quotetheorem:80] [citeproof:80] An important remark on the continuous-versus-simplicial distinction is necessary here. Because we work with simplicial homology, one might worry that the argument only shows simplicial approximations of $f$ have fixed points, not $f$ itself. The passage from simplicial approximations to $f$ is non-immediate — it is precisely because homology is a homotopy invariant, and because $f$ and any of its simplicial approximations are homotopic (hence induce the same maps on homology), that the conclusion applies to $f$ directly. [illustration:brouwer-retraction-construction] ### The Antipodal Map and the Hairy Ball Theorem Having computed the homology of spheres, we can ask: when is the antipodal map $a: S^n \to S^n$, $x \mapsto -x$, homotopic to the identity? The answer depends entirely on the parity of $n$, and homology gives the definitive obstruction. Recall that for odd $n$, the antipodal map is homotopic to the identity — this was an exercise on the first example sheet, constructed by rotating $S^n$ viewed as a product of circles in $\mathbb{R}^{n+1}$. For even $n$, this rotation trick fails, and we will now show it genuinely cannot work. The strategy is to compute $a_*: H_n(S^n) \to H_n(S^n)$, which is a map $\mathbb{Z} \to \mathbb{Z}$, hence multiplication by some integer. If $a \simeq \operatorname{id}$ then $a_* = \operatorname{id}_* = \operatorname{id}$, i.e.\ multiplication by $+1$. We will show $a_*$ is multiplication by $(-1)^{n+1}$, which equals $-1$ when $n$ is even. To compute $a_*$, we need a triangulation of $S^n$ that is compatible with $a$. The standard triangulation $\partial \Delta^{n+1}$ is not compatible with the antipodal map, since $a$ does not permute its vertices. Instead, we use the triangulation with vertex set \begin{align*} V_K = \{\pm e_0, \pm e_1, \ldots, \pm e_n\} \subset \mathbb{R}^{n+1}, \end{align*} where $e_i$ are the standard basis vectors. The antipodal map sends $e_i \mapsto -e_i$, so it permutes vertices and is simplicial in this triangulation. [quotetheorem:1948] [citeproof:1948] With this generator in hand, the antipodal map acts by $a_*x = (-1)^{n+1} x$, since applying $e_i \mapsto -e_i$ to the oriented simplex $(\varepsilon_0 e_0, \ldots, \varepsilon_n e_n)$ introduces a sign flip in each of the $n+1$ coordinates, contributing a factor of $(-1)^{n+1}$ to the overall coefficient. [quotetheorem:1950] [citeproof:1950] This result is the algebraic core of the **hairy ball theorem**: there is no nowhere-vanishing continuous tangent vector field on $S^n$ when $n$ is even. The argument runs as follows. Given a nowhere-vanishing vector field $v$ on $S^n$, we can normalise it to get a continuous map $S^n \to S^n$ assigning to each point a unit tangent vector. One then constructs a homotopy from the identity to the antipodal map via $H(x,t) = \cos(\pi t)\, x + \sin(\pi t)\, v(x)$, which rotates each point along the great circle in the direction of $v(x)$. If $v$ existed on $S^n$ with $n$ even, the antipodal map would be homotopic to the identity — contradicting what we just proved. ## Homology of Surfaces Computing the homology groups of compact surfaces is both a showcase of the Mayer-Vietoris sequence in action and a connection back to the classical classification theorem. The goal is to see how the genus $g$ is perfectly captured by the rank of $H_1$. We assume all compact surfaces are triangulable — this is a theorem in surface topology that we do not prove here, but it holds for all surfaces arising from the classification. The classification of compact orientable surfaces gives the oriented surfaces $\Sigma_g$ for $g \geq 0$: the sphere $\Sigma_0 = S^2$, the torus $\Sigma_1$, and the genus-$g$ surface for general $g$. The non-orientable surfaces are the connected sums $E_n$ of $n$ copies of $\mathbb{RP}^2$, beginning with $E_1 = \mathbb{RP}^2$ and $E_2 = $ the Klein bottle. [illustration:orientable-surfaces-genus] ### Computing Homology of Oriented Surfaces via Mayer-Vietoris The direct approach — writing down a triangulation of $\Sigma_g$ and computing the chain complex — is feasible but laborious. The Mayer-Vietoris sequence gives a slicker path by decomposing $\Sigma_g$ into simpler pieces. The key decomposition comes from the polygon model. Recall that $\Sigma_g$ is obtained from a $4g$-gon by gluing edge pairs according to the word $a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1}$. The surface $F_g$ is defined as $\Sigma_g$ with an open disk removed — topologically, a compact surface with one boundary circle. We decompose: \begin{align*} \Sigma_g = F_{g-1} \cup F_1, \end{align*} where the two pieces are glued along their common boundary circle $S^1$. [illustration:surface-cut-along-circle] To use Mayer-Vietoris, we need to know the homology of $F_g$. This follows from a deformation retraction: starting from the polygon model of $\Sigma_g$ and removing a disk from the center, one expands the removed disk outward until it reaches the boundary. The resulting space is the boundary polygon with its edges glued — which is a wedge of $2g$ circles, the rose $X_{2g}$. [quotetheorem:1952] The deformation retraction $F_g \simeq X_{2g}$ is geometric: radially push the punctured polygon out to its boundary, and the boundary with edges identified is precisely the $2g$-petal rose. The homology of $X_{2g}$ follows from Mayer-Vietoris applied iteratively to wedge summands: each additional circle contributes a $\mathbb{Z}$ to $H_1$ and nothing in higher degrees. With these ingredients assembled, the Mayer-Vietoris sequence for $\Sigma_g = F_{g-1} \cup_{S^1} F_1$ reads (with intersection $S^1$): \begin{align*} 0 \to H_2(S^1) \to H_2(F_{g-1}) \oplus H_2(F_1) \to H_2(\Sigma_g) \to H_1(S^1) \to H_1(F_{g-1}) \oplus H_1(F_1) \to \cdots \end{align*} Since $F_{g-1}$ and $F_1$ have no $H_2$ (they are homotopy equivalent to graphs) and $H_2(S^1) = 0$, the sequence simplifies in the top degree. Substituting known groups: \begin{align*} 0 \to H_2(\Sigma_g) \xrightarrow{\partial} \mathbb{Z} \xrightarrow{\phi} \mathbb{Z}^{2g} \to H_1(\Sigma_g) \to \mathbb{Z} \xrightarrow{\psi} \mathbb{Z}^{2g-2} \oplus \mathbb{Z}^2 \to \mathbb{Z} \to 0. \end{align*} The map $\phi: H_1(S^1) \to H_1(F_{g-1}) \oplus H_1(F_1)$ sends the generator of $H_1(S^1)$ to the boundary circle of each piece. After the deformation retraction $F_g \simeq X_{2g}$, this boundary circle maps to the boundary of the polygon, which traverses each $1$-cell twice — once in each orientation. These contributions cancel, so $\phi = 0$. Since $\phi = 0$, exactness gives $H_2(\Sigma_g) \cong \ker \phi = \mathbb{Z}$. This is expected: the oriented surface $\Sigma_g$ is a closed orientable manifold, so its top homology is $\mathbb{Z}$, generated by the fundamental class. For $H_1(\Sigma_g)$, we extract a short exact sequence from the long exact sequence. The map $\psi: H_0(S^1) \to H_0(F_{g-1}) \oplus H_0(F_1)$ sends $1 \mapsto (1,1)$ (since the circle maps to a point in each connected piece), so $\psi$ is injective and $\ker \psi = 0$. Therefore the long exact sequence breaks as: \begin{align*} 0 \to \mathbb{Z}^{2g} \to H_1(\Sigma_g) \to 0, \end{align*} giving $H_1(\Sigma_g) \cong \mathbb{Z}^{2g}$. [quotetheorem:1954] [citeproof:1954] The result is a beautiful confirmation of the classification theorem: the genus $g$ is read off directly from the rank of the first homology group, $\operatorname{rank}(H_1(\Sigma_g)) = 2g$. Two surfaces are homeomorphic if and only if they have the same genus, and this is reflected precisely in their homology. The second homology $H_2(\Sigma_g) \cong \mathbb{Z}$ for all $g \geq 0$ captures the orientability — it says each surface carries a fundamental class, the algebraic shadow of a consistent orientation. This computation is paradigmatic of how Mayer-Vietoris is used in practice. One writes down the long exact sequence, inserts the known groups, and then identifies one or two of the maps to pin down the remaining unknowns. The art is in understanding those maps geometrically — here, recognising that the boundary circle traverses each edge with opposite orientations is the key step. ### Homology of Non-Orientable Surfaces For the non-orientable surfaces $E_n$ (the connected sum of $n$ copies of $\mathbb{RP}^2$), the integral homology groups acquire a torsion part: \begin{align*} H_k(E_n) \cong \begin{cases} \mathbb{Z} & k = 0, \\ \mathbb{Z}^{n-1} \oplus \mathbb{Z}/2 & k = 1, \\ 0 & k \geq 2. \end{cases} \end{align*} The $\mathbb{Z}/2$ term in $H_1$ is characteristic of non-orientability: the Möbius band in $\mathbb{RP}^2$ gives rise to a class of order $2$, and connected summing adds free generators. Notably, $H_2(E_n) = 0$ — non-orientable surfaces carry no fundamental class over $\mathbb{Z}$, which is the algebraic expression of their non-orientability. The contrast with the orientable case is sharp. For $\Sigma_g$, the first homology is entirely free of rank $2g$, the second homology is $\mathbb{Z}$, and there is no torsion. For $E_n$, the second homology vanishes and the first homology has a $\mathbb{Z}/2$ torsion piece, plus $n-1$ free generators. Homology thus perfectly distinguishes the two families. ## Rational Homology, Euler and Lefschetz Numbers Working with integral homology groups captures the full topological information, but at the cost of handling finitely generated abelian groups with their torsion subgroups. Replacing $\mathbb{Z}$ with $\mathbb{Q}$ as the coefficient ring turns homology groups into vector spaces, making the linear-algebraic machinery of dimension, trace, and rank directly available — but at the price of losing torsion information. [definition: Rational Chain Group and Rational Homology] For a simplicial complex $K$, the **rational $n$-chain group** $C_n(K; \mathbb{Q})$ is the vector space over $\mathbb{Q}$ with basis the oriented $n$-simplices of $K$ (with a fixed choice of orientation for each). The boundary maps $d_n: C_n(K; \mathbb{Q}) \to C_{n-1}(K; \mathbb{Q})$ are defined by the same formula as in the integral case. The **rational $n$th homology group** is \begin{align*} H_n(K; \mathbb{Q}) := \frac{Z_n(K; \mathbb{Q})}{B_n(K; \mathbb{Q})}, \end{align*} where $Z_n$ and $B_n$ denote rational cycles and boundaries respectively. [/definition] Since $\mathbb{Q}$ is a field, each $C_n(K; \mathbb{Q})$ is a finite-dimensional vector space, and the homology groups $H_n(K; \mathbb{Q})$ are also finite-dimensional vector spaces. This means we can speak of their dimensions, denoted $b_n(K) = \dim_\mathbb{Q} H_n(K; \mathbb{Q})$, which are called the **Betti numbers**. The relationship between rational and integral homology is clean and explicit. [quotetheorem:1955] The proof follows from the universal coefficient theorem for homology, or directly from the observation that tensoring $\mathbb{Z}^k \oplus F$ with $\mathbb{Q}$ kills the finite part ($F \otimes_\mathbb{Z} \mathbb{Q} = 0$ since every element of $F$ has finite order, so is annihilated by tensoring with $\mathbb{Q}$) and converts the free part to $\mathbb{Q}^k$. Passing to rational homology thus amounts to discarding all torsion information. In good cases — such as the orientable surfaces $\Sigma_g$ — no information is lost, since integral homology is already torsion-free. In other cases, significant information disappears. [example: Rational Homology of Spheres, Surfaces, and Projective Plane] For spheres, rational homology retains all information: \begin{align*} H_k(S^n; \mathbb{Q}) \cong \begin{cases} \mathbb{Q} & k = 0 \text{ or } k = n, \\ 0 & \text{otherwise.} \end{cases} \end{align*} For the orientable surface $\Sigma_g$, the integral homology has no torsion, so: \begin{align*} H_k(\Sigma_g; \mathbb{Q}) \cong \begin{cases} \mathbb{Q} & k = 0 \text{ or } k = 2, \\ \mathbb{Q}^{2g} & k = 1, \\ 0 & \text{otherwise.} \end{cases} \end{align*} For the non-orientable surface $E_n$, the $\mathbb{Z}/2$ torsion in $H_1$ disappears: \begin{align*} H_k(E_n; \mathbb{Q}) \cong \begin{cases} \mathbb{Q} & k = 0, \\ \mathbb{Q}^{n-1} & k = 1, \\ 0 & \text{otherwise.} \end{cases} \end{align*} Here $E_n$ and $\Sigma_g$ differ in their rational $H_1$ (unless $n-1 = 2g$ by coincidence), but the integral $H_1(E_n)$ has an additional $\mathbb{Z}/2$ that rational homology does not see. For $\mathbb{RP}^2$, since $H_1(\mathbb{RP}^2) \cong \mathbb{Z}/2$ and $H_k(\mathbb{RP}^2) = 0$ for $k \geq 2$, all non-zero homology above degree zero is torsion. Thus $H_n(\mathbb{RP}^2; \mathbb{Q}) \cong H_n(\text{pt}; \mathbb{Q})$ for all $n$ — over $\mathbb{Q}$, the projective plane is indistinguishable from a point. [/example] The fact that $\mathbb{RP}^2$ looks like a point over $\mathbb{Q}$ is a striking illustration of the information lost by rationalisation. However, the advantage gained — working with vector spaces and linear algebra — is substantial, as the Euler characteristic and Lefschetz number show. ### The Euler Characteristic The classical Euler characteristic $V - E + F$ (vertices minus edges plus faces) is the oldest topological invariant of surfaces. But defined this way, it depends on the triangulation, and extending it to higher dimensions by counting cells of each dimension still leaves open whether the result is a topological invariant. Rational homology resolves both issues at once. [definition: Euler Characteristic] The **Euler characteristic** of a triangulable space $X$ is \begin{align*} \chi(X) = \sum_{i \geq 0} (-1)^i \dim_\mathbb{Q} H_i(X; \mathbb{Q}). \end{align*} [/definition] This definition makes the topological invariance manifest: since homology groups depend only on the homotopy type of $X$, so does $\chi(X)$. Homotopy equivalent spaces have equal Euler characteristics. We will verify below that this definition recovers the classical cell-counting formula. [example: Euler Characteristics of Familiar Spaces] For spheres: \begin{align*} \chi(S^n) = \begin{cases} 2 & n \text{ even}, \\ 0 & n \text{ odd.} \end{cases} \end{align*} This is because $\dim H_0 = \dim H_n = 1$ and all other Betti numbers vanish, giving $\chi = 1 + (-1)^n$, which is $2$ for even $n$ and $0$ for odd $n$. For oriented surfaces, substituting the Betti numbers $b_0 = b_2 = 1$ and $b_1 = 2g$: \begin{align*} \chi(\Sigma_g) = 1 - 2g + 1 = 2 - 2g. \end{align*} For non-orientable surfaces, with $b_0 = 1$, $b_1 = n-1$, $b_2 = 0$: \begin{align*} \chi(E_n) = 1 - (n-1) = 2 - n. \end{align*} [/example] These formulas are consistent with what one can compute directly from triangulations, confirming that the two definitions agree. The genus formula $\chi(\Sigma_g) = 2 - 2g$ is famous: it says the sphere has $\chi = 2$, the torus has $\chi = 0$, and each additional handle reduces the Euler characteristic by $2$. ### The Lefschetz Number The Euler characteristic is the Lefschetz number of the identity map. The Lefschetz number generalises this by replacing the identity with an arbitrary continuous self-map, producing a single integer that encodes how the map deforms the space. [definition: Lefschetz Number] Given a continuous map $f: X \to X$ where $X$ is a triangulable space, the **Lefschetz number** of $f$ is \begin{align*} L(f) = \sum_{i \geq 0} (-1)^i \operatorname{tr}(f_*: H_i(X; \mathbb{Q}) \to H_i(X; \mathbb{Q})). \end{align*} [/definition] The trace is the linear-algebraic trace of the map $f_*$ viewed as a linear endomorphism of the finite-dimensional $\mathbb{Q}$-vector space $H_i(X; \mathbb{Q})$. Since $\operatorname{tr}(\operatorname{id}_{V}) = \dim V$, taking $f = \operatorname{id}$ recovers $\chi(X) = L(\operatorname{id})$, so the Euler characteristic is indeed a special case. [example: Lefschetz Number of the Antipodal Map] For the antipodal map $a: S^n \to S^n$, we computed $a_*: H_n(S^n; \mathbb{Q}) \to H_n(S^n; \mathbb{Q})$ is multiplication by $(-1)^{n+1}$. In degree $0$, the antipodal map fixes the connected component, so $a_*: H_0(S^n; \mathbb{Q}) \to H_0(S^n; \mathbb{Q})$ is the identity, with trace $1$. Therefore: \begin{align*} L(a) = 1 + (-1)^n \cdot (-1)^{n+1} = 1 + (-1)^{2n+1} = 1 - 1 = 0. \end{align*} The Lefschetz number of the antipodal map is $0$ for every $n$, regardless of the dimension. This is not a coincidence — the antipodal map never has a fixed point, and the Lefschetz fixed point theorem, which we prove next, guarantees $L(f) = 0$ whenever $f$ is fixed-point free. [/example] ### The Trace Formula for Chain Maps To apply the Lefschetz number computationally, it is important to understand it in terms of chain groups rather than homology groups. The chain groups are directly determined by the triangulation and are easier to work with concretely. The following lemma is the key linear-algebraic step. [quotetheorem:1957] [citeproof:1957] This lemma allows us to replace the maps on homology groups with maps on chain groups when computing the Lefschetz number. The homology groups sit as quotients of the cycles, which in turn sit inside the chain groups — so two applications of the lemma decomposes the trace on each chain group into contributions from boundaries, cycles, and homology. [quotetheorem:1958] [citeproof:1958] This formula also immediately recovers the classical Euler characteristic: taking $f = \operatorname{id}$, the trace of the identity on $C_i(K; \mathbb{Q})$ is $\dim C_i(K; \mathbb{Q})$, which equals the number of $i$-simplices. Thus: \begin{align*} \chi(X) = \sum_{i \geq 0} (-1)^i \cdot (\text{number of $i$-simplices of } K), \end{align*} confirming that the homological definition of $\chi$ agrees with the classical cell-counting formula and in particular is independent of the triangulation. ### The Lefschetz Fixed Point Theorem The Lefschetz fixed point theorem is the promised generalisation of Brouwer's theorem. Instead of requiring $X$ to be contractible (like $D^n$), it applies to any triangulable space, with the Lefschetz number serving as the fixed-point detector. [quotetheorem:1959] [citeproof:1959] The proof reveals the geometric mechanism: the Lefschetz number is computed by tracing how each simplex maps to other simplices, and if $f$ is fixed-point free then $f$ moves every simplex off itself, forcing every diagonal trace to vanish. The two standard examples illustrate the theorem's scope beautifully. [example: Contractible Spaces Have Fixed Points for All Maps] Let $X$ be any contractible triangulable polyhedron, such as $D^n$. Then $H_0(X; \mathbb{Q}) \cong \mathbb{Q}$ and $H_i(X; \mathbb{Q}) = 0$ for $i > 0$. For any $f: X \to X$, the induced map $f_*: H_0(X; \mathbb{Q}) \to H_0(X; \mathbb{Q})$ is the identity (since $X$ is path-connected, all points lie in the same component, and $f_*$ acts as the identity on $H_0$). Therefore $L(f) = \operatorname{tr}(\operatorname{id}_\mathbb{Q}) = 1 \neq 0$, so $f$ must have a fixed point. This recovers Brouwer's theorem for all contractible polyhedra. [/example] [example: Topological Groups Have Euler Characteristic Zero] Suppose $G$ is a path-connected topological group — a group whose underlying space is a topological space and whose multiplication and inversion are continuous. If $g \in G$ is any element other than the identity, the right-multiplication map $r_g: G \to G$, $\gamma \mapsto \gamma g$, has no fixed point (since $\gamma g = \gamma$ implies $g = e$). The Lefschetz fixed point theorem forces $L(r_g) = 0$. On the other hand, since $G$ is path-connected, any element $g$ can be connected to the identity $e$ by a path. Right-multiplying along this path provides a homotopy from $r_g$ to $r_e = \operatorname{id}_G$. Since $L$ is a homotopy invariant: \begin{align*} \chi(G) = L(\operatorname{id}_G) = L(r_g) = 0. \end{align*} So any non-trivial path-connected topological group has Euler characteristic zero. Among the surfaces, $\chi(\Sigma_g) = 2 - 2g = 0$ only for $g = 1$, i.e. the torus $\Sigma_1 = S^1 \times S^1$. Indeed, the torus is a topological group (componentwise multiplication in $S^1 \times S^1$). The sphere $\Sigma_0 = S^2$ has $\chi = 2$, so it cannot carry a topological group structure. Similarly, $\chi(\Sigma_g) \neq 0$ for $g \geq 2$, ruling out those surfaces as well. The Klein bottle $E_2$ has $\chi = 0$ and is a candidate — and indeed it does carry a topological group structure, though its construction is more involved. [/example] The Lefschetz fixed point theorem thus provides a bridge between global topology (Euler characteristic, Betti numbers) and local dynamics (fixed points of maps). It is one of the crowning results of simplicial homology, and its proof perfectly captures the central philosophy of the subject: translate geometric questions into chain-level linear algebra, where the answer becomes a matter of computing traces. ## References - Wilton, H. (2015). *Algebraic Topology*. Lecture notes, Part IB, Michaelmas 2015, University of Cambridge.