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.