Algebraic Topology

Citation

Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. Free official edition: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html

Chapter Summary

Chapter 0: Some Underlying Geometric Notions

This short preliminary chapter introduces, informally and mostly without proofs, the geometric ideas used throughout the book: homotopy, deformation retractions, mapping cylinders and homotopy equivalence; CW (cell) complexes built by attaching cells; operations on spaces such as products, quotients, suspensions, joins and wedge sums; two criteria for homotopy equivalence; and the homotopy extension property. The author says it can be read first or skipped and consulted later as needed.

Chapter 1: The Fundamental Group

Motivated by examples of linked and unlinked circles, the chapter defines paths, homotopy and the fundamental group, computes π₁(S¹) ≅ ℤ, and draws early consequences: the fundamental theorem of algebra, the Brouwer fixed point theorem in dimension 2 and the Borsuk–Ulam theorem for S². Section 1.2 introduces free products and van Kampen's theorem and uses them to compute fundamental groups of cell complexes from their 2-skeleta. Section 1.3 develops covering spaces—lifting properties, the classification of covering spaces by subgroups of π₁, deck transformations and group actions. Additional topics treat graphs and free groups (including subgroups of free groups) and K(G,1) spaces and graphs of groups.

Chapter 2: Homology

After explaining why higher homotopy groups are hard to compute and homology is more tractable, the chapter builds homology in stages: Δ-complexes and simplicial homology, then singular homology, with homotopy invariance, long exact sequences, excision and the equivalence of simplicial and singular homology. Early applications include Brouwer's fixed point theorem and invariance of dimension. Section 2.2 develops computational tools—degree of maps of spheres, cellular homology, Euler characteristic, Mayer–Vietoris sequences and homology with coefficients—and proves that Sⁿ has a nonvanishing tangent vector field exactly when n is odd. Section 2.3 presents the axioms for homology and the language of categories and functors. Additional topics cover the relation between H₁ and π₁, classical applications (the Jordan curve theorem, invariance of domain, division algebras, the Borsuk–Ulam theorem) and simplicial approximation with the Lefschetz fixed point theorem.

Chapter 3: Cohomology

Cohomology is introduced as the dual of homology, with the universal coefficient theorem showing how one determines the other. The chapter's emphasis is the extra structure that contravariance gives: the cup product making cohomology a ring, a Künneth formula, and computations of the cohomology rings of projective spaces, with an application to division algebra structures on ℝⁿ. Section 3.3 develops orientations of manifolds and Poincaré duality, its connection with cup products, and other forms of duality such as Alexander duality. The extensive additional topics include universal coefficients for homology, the general Künneth formula, H-spaces and Hopf algebras, the cohomology of SO(n), Bockstein homomorphisms, limits and Ext, transfer homomorphisms and local coefficients.

Chapter 4: Homotopy Theory

Higher homotopy groups are defined with their long exact sequence, followed by Whitehead's theorem, cellular approximation and CW approximation. Section 4.2 covers excision for homotopy groups (with the Freudenthal suspension theorem), the Hurewicz theorem, fiber bundles and their long exact sequence, and stable homotopy groups, including tables of known stable homotopy groups of spheres. Section 4.3 connects homotopy with cohomology through Eilenberg–MacLane spaces, representing cohomology classes by maps into K(G, n), and introduces fibrations, Postnikov towers and obstruction theory. Twelve additional topics range from basepoint issues and the Hopf invariant to the cohomology of fiber bundles (Leray–Hirsch, Gysin and Thom isomorphism), Brown representability, spectra and homology theories, the Dold–Thom theorem and Steenrod squares and powers.

Appendix

The appendix collects point-set topological material that the main text treats lightly: the topology of cell complexes, the compact-open topology, the homotopy extension property and simplicial CW structures.

Key Concepts

Chapter 0: Some Underlying Geometric Notions

• Homotopy equivalence: maps f: X → Y and g: Y → X whose composites are homotopic to the identities, so X and Y have "the same shape".
• CW complex: a space built inductively by attaching cells of increasing dimension, the main class of spaces in the book.
• Homotopy extension property: the ability to extend a homotopy defined on a subspace to the whole space, which holds for CW pairs.

Chapter 1: The Fundamental Group

• Fundamental group π₁(X, x₀): the group of homotopy classes of loops at a basepoint, with composition of loops as the operation.
• Van Kampen's theorem: computes the fundamental group of a union of open sets from the fundamental groups of the pieces and their intersections.
• Covering space: a map that is locally a product with a discrete fiber; connected coverings of a nice space correspond to conjugacy classes of subgroups of π₁.

Chapter 2: Homology

• Singular homology: homology built from all continuous maps of simplices into a space, homotopy invariant and satisfying excision.
• Long exact sequence and excision: tools relating the homology of a pair, a subspace and a quotient, the basis of most computations.
• Cellular homology: an efficient method computing homology of a CW complex from a chain complex with one generator per cell.
• Mayer–Vietoris sequence: the homology analogue of van Kampen's theorem for a space decomposed into two pieces.

Chapter 3: Cohomology

• Universal coefficient theorem: expresses cohomology in terms of homology via Hom and Ext.
• Cup product: a product making H*(X; R) a graded ring, finer than the additive structure.
• Poincaré duality: for a closed orientable n-manifold, H^k and H_{n−k} are isomorphic, with cup product giving a nonsingular pairing.

Chapter 4: Homotopy Theory

• Whitehead's theorem: a map of CW complexes inducing isomorphisms on all homotopy groups is a homotopy equivalence.
• Hurewicz theorem: for an (n−1)-connected space, the first nonzero homotopy group is isomorphic to the first nonzero homology group.
• Fiber bundle and fibration: maps with a homotopy lifting property that give long exact sequences of homotopy groups.
• Eilenberg–MacLane space K(G, n): a space with a single nonzero homotopy group, representing cohomology Hⁿ(–; G).

Critical Analysis

Strengths

• Geometric, readable exposition. The preface states that readability is a first priority and that the book leans towards geometric rather than algebraic aspects. Many pictures and worked examples accompany the arguments, from linked circles in Chapter 1 to cell structures on projective spaces and lens spaces.
• CW complexes and Δ-complexes make computation efficient. CW complexes are emphasised as the natural class of spaces, and the Δ-complex structure (for example, a torus from 2 triangles instead of the 14 needed for a simplicial complex) simplifies early homology computations.
• Core plus substantial optional material. Each chapter has a core sequence followed by additional topics that together make up nearly half the book, so the text serves both a first course and later reference, and readers can choose the order of homology, cohomology and homotopy.
• Many exercises. Every section and additional topic ends with an exercise set, ranging from routine verifications to substantial problems, a few with hints or references to the literature.

Limitations

• Graduate-level prerequisites. The preface assumes undergraduate algebra and point-set topology, especially quotient spaces; point-set details are often treated lightly in the text and left to the appendix, which can trouble readers without that background.
• No spectral sequences or K-theory. Spectral sequences are deliberately postponed to a separate volume, and vector bundles, characteristic classes and K-theory to another, so some standard computations are outside this book.
• No solutions and uneven difficulty. The book provides no answers to exercises, and the additional topics vary considerably in difficulty, which makes self-study demanding.
• Classical in scope. The author describes the material as essentially classical; modern developments such as model categories or higher category theory are not discussed.

Applications in the Book

Applications are concentrated in early sections and in the additional topics: the fundamental theorem of algebra, the Brouwer fixed point theorem and the Borsuk–Ulam theorem (Section 1.1); invariance of dimension, the hairy ball theorem on tangent vector fields on spheres and the Euler characteristic (Chapter 2); the Jordan curve theorem, invariance of domain, division algebras and the Lefschetz fixed point theorem (Sections 2.B–2.C); the restriction of division algebra structures on ℝⁿ via cup products (Section 3.2); and Poincaré and Alexander duality for manifolds and complements of subspaces (Section 3.3).

Where It Fits in the Course

In this Topology and Geometry list, Hatcher is the graduate-level algebraic topology text. Richard Earl's Topology: A Very Short Introduction and Jeffrey Weeks's The Shape of Space are generally described as informal, intuitive introductions to topological ideas and the geometry of surfaces and 3-manifolds; they are useful background reading but not preparation for the proofs here. Sutherland's Introduction to Metric and Topological Spaces is generally described as a first course in point-set topology and supplies the quotient spaces, compactness and connectedness that Hatcher assumes.

Munkres's Topology covers point-set topology and, in its second part, an introduction to the fundamental group and covering spaces; studying it, or Sutherland, before Hatcher gives the needed foundation, and its algebraic topology chapters overlap with Hatcher's Chapter 1. Hatcher is the book to read after these, for homology, cohomology and higher homotopy theory.

How to Use This Book

The book is aimed at graduate students and advanced undergraduates. Readers should know group theory and basic ring and module theory (for coefficients and cup products), and point-set topology including quotient spaces; Chapter 0 and the appendix supply geometric and point-set background.

A typical first course covers Chapter 0 as needed, Chapter 1 (Sections 1.1–1.3) and Chapter 2 (Sections 2.1–2.2), possibly with 2.B and parts of Chapter 3 (cohomology, cup products and Poincaré duality). A second course continues with Chapter 3 and the core of Chapter 4 (Sections 4.1–4.3). The additional topics can be sampled according to interest—for example 1.A for free groups, 2.B for classical applications, 4.D for fiber bundles or 4.L for Steenrod squares. Work through the examples carefully with pictures, attempt the exercises at the end of each section, and check the author's website for the errata list and supplementary material.

Edition and Sources

This guide is based on the free electronic edition of Algebraic Topology on the author's Cornell website (PDF dated October 2022, incorporating the corrections of the 2015 reprinting of the Cambridge University Press edition first published in 2002), https://pi.math.cornell.edu/~hatcher/AT/ATpage.html. It draws on the preface (aims, structure, prerequisites, Δ-complexes, companion volumes and the 2015 reprinting note), the standard notations, the full table of contents including all additional topics and the appendix, the introductions to all five chapters, every section and subsection heading, the statements of the main numbered theorems in Chapters 1–4, and sample exercise sets.