Algebraic topology is the art of assigning algebraic invariants to topological spaces in a way that respects continuous deformations. Rather than studying the intricate details of how a space is constructed geometrically, we instead ask: what algebraic information can we extract that remains constant whenever the space is deformed continuously? This philosophy transforms topology from a study of shapes into a study of algebraic structures like groups, rings, and modules — tools far more amenable to computation and reasoning.

The course begins with homotopy, the formalization of continuous deformation, which provides both the conceptual foundation and the key mechanism ensuring that our invariants are well-defined: two homotopic maps induce the same algebraic effects. From there, we develop singular homology, arguably the most natural and powerful such invariant, by counting and canceling certain algebraic combinations of simplices. The early chapters — covering singular (co)homology, the four major computational tools, and reduced homology — establish the machinery needed to compute homology groups for concrete spaces. Cell complexes provide a dramatically more efficient framework for these calculations, reducing the infinite-dimensional chain complexes of singular homology to finite computations.

The later chapters expand both the scope and power of homology theory. We generalize to arbitrary coefficient rings rather than just integers, introducing cohomology as the dual perspective and revealing hidden multiplicative structure through the cup product. The Kunneth theorem and universal coefficients theorem clarify how homology behaves under products and extensions, while vector bundles and manifolds showcase applications to high-dimensional geometry. Poincare duality stands as the capstone: a profound symmetry asserting that the homology and cohomology of a closed oriented manifold are dual to each other, revealing an elegant interplay between geometry and algebra that has shaped topology for over a century.

# Introduction

This chapter introduces the central organizing idea of algebraic topology: studying spaces and maps up to continuous deformation. Throughout the course, the word "map" means "continuous function." We define homotopy between maps, homotopy equivalence between spaces, and explain why these are the natural notions of sameness when one wants invariants that see topology rather than geometry. The chapter closes by motivating the algebraic machinery — homology and cohomology — that the rest of the course develops.

## Homotopy Between Maps

Topology studies spaces up to homeomorphism, but homeomorphism is a very rigid notion: it asks for a bijective continuous map with a continuous inverse. In practice we often have two maps $f_0, f_1: X \to Y$ that are "morally the same" — they describe paths in the same homology class, or sections over the same base — yet are not literally equal. Treating them as genuinely different would make every algebraic invariant hopelessly sensitive to irrelevant perturbations. What we need is a way to declare two maps equivalent whenever one can be continuously deformed into the other. Without such a notion, the algebraic objects we attach to spaces would not be invariants at all: a tiny deformation of a map could change the computed homology class, and we would have no stable theory.

To see what goes wrong without homotopy, consider two maps from $[0,1]$ into $\mathbb{R}^2 \setminus \{0\}$: the path $f_0(t) = (\cos \pi t, \sin \pi t)$ (a semicircle above the origin) and $f_1(t) = (\cos \pi t, -\sin \pi t)$ (a semicircle below). Both run from $(-1, 0)$ to $(1, 0)$, and they are not equal as maps. But topologically they behave identically with respect to every invariant that does not see the hole at the origin — there is no obstruction to deforming one into the other within $\mathbb{R}^2 \setminus \{0\}$ while keeping the endpoints fixed. If our theory distinguished these two maps, it would be measuring an irrelevant geometric accident rather than genuine topology. The correct framework identifies them as equivalent, which motivates the following definition.

[definition: Homotopy] Let $X$ and $Y$ be topological spaces. A **homotopy** between maps $f_0, f_1: X \to Y$ is a map $F: [0,1] \times X \to Y$ such that \begin{align*} F(0, x) = f_0(x) \quad \text{and} \quad F(1, x) = f_1(x) \end{align*} for all $x \in X$. If such an $F$ exists, we say $f_0$ is **homotopic** to $f_1$ and write $f_0 \simeq f_1$. [/definition]

Think of $t \in [0,1]$ as a time parameter. At time $t = 0$ we see the map $f_0$; at time $t = 1$ we see the map $f_1$; and for intermediate $t$ we see a continuous family of maps interpolating between them.

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The key feature of homotopic maps is that they "do the same thing" from the perspective of the algebraic invariants this course develops. This will be made precise by the homotopy invariance theorem in Chapter 3.

## Homotopy Equivalence

If homotopic maps are to be regarded as the same, we need a correspondingly enlarged notion of isomorphism between spaces. The idea is to replace equality in the definition of homeomorphism by homotopy.

[definition: Homotopy Equivalence] A map $f: X \to Y$ is a **homotopy equivalence** if there exists a map $g: Y \to X$ such that \begin{align*} f \circ g \simeq \operatorname{id}_Y \quad \text{and} \quad g \circ f \simeq \operatorname{id}_X. \end{align*} The map $g$ is called a **homotopy inverse** to $f$. When a homotopy equivalence exists we write $X \simeq Y$ and say $X$ and $Y$ are **homotopy equivalent** (or have the same **homotopy type**). [/definition]

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This transitivity argument is the key place where the Composition of Homotopies theorem is used. This relies on the fact that composition is compatible with homotopy.

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