Algebraic topology attaches computable algebraic objects to spaces while preserving the effect of continuous maps. Hatcher develops fundamental groups and homology with geometric constructions and exercises. May gives a more compact graduate bridge toward homotopy theory. Start with Hatcher when loops, attaching maps and cellular calculations are new; use May when you can already supply those examples while reading a compressed argument.
The ambient space matters
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Annulus
Algebraic Topology — Allen Hatcher. Start with chapter 0 on homotopy and cell complexes. Read 1.1 Basic Constructions, 1.2 Van Kampen's Theorem and 1.3 Covering Spaces; then 2.1 Simplicial and Singular Homology and the Cellular Homology and Mayer-Vietoris subsections of 2.2.
Choose a space built from a small number of cells and calculate the same invariant in two ways where possible. In van Kampen, specify the open cover and basepoint information before manipulating generators. For cellular homology, write the boundary maps and check their compositions. Record the map induced by a continuous map, rather than stopping at the groups of the two spaces. Compare homotopy equivalence with equality of homology groups so the invariant's limitations remain visible.
Learn fundamental groups, covering spaces and cellular homology with explicit spaces.
Start with chapter 0 on homotopy and cell complexes. Read 1.1 Basic Constructions, 1.2 Van Kampen's Theorem and 1.3 Covering Spaces; then 2.1 Simplicial and Singular Homology and the Cellular Homology and Mayer-Vietoris subsections of 2.2.
Source · Checked 2026-10-16
Revisit foundational constructions after working concrete calculations.
Use the fundamental constructions as a graduate bridge, following the revised PDF contents; chapter numbers were not freshly checked.
Source · Checked 2026-10-11
Fundamental-group and homology exercises. Exercise sets at the ends of sections 1.1-1.3 and 2.1-2.2. Start with 1.1 exercises 10, 12 and 14 for products and induced maps; use 2.2 for cellular computations.. Public exercises verified in the book; an author-provided public solution collection was not verified.
Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.
Using the usual CW structure with one cell in each dimension 0, 1 and 2, compute the integral homology of the real projective plane, including the degree-2 cellular boundary map.
The 1-skeleton is a circle. The attaching map of the 2-cell traverses it twice, while the 1-cell has the same initial and terminal vertex.
Use the disk model of with antipodal boundary points identified. The boundary quotient is , its -skeleton. The attaching map from the boundary of the disk to this circle is the double covering; with compatible orientations it has degree . Thus the cellular chain complex over is
The degree-one boundary is zero because the single -cell has both endpoints at the same -cell. Therefore , , and . All higher groups vanish because there are no higher cells. Reversing a cell orientation can change the displayed to but changes neither its kernel nor its image. In particular, the -cell does not supply a nonzero integral top-dimensional homology class.
If this was the sticking point:
Already comfortable with this? Skip fundamental-group review if you can carry out van Kampen with valid open sets and track induced maps. Skip the homology selection if you can compute the projective-plane example with integer coefficients and explain why the attaching-map degree matters.
Review the prerequisites · Compare the full subject collection
Geometric examples accompany fundamental groups, homology and cohomology. A substantial first graduate text; 'introductory' refers to the field, not to university mathematics.
2000 archive · noncommercial paper or electronic copies permitted; other rights reserved.
Develops homotopy-theoretic constructions in a categorical setting. A continuation after basic topology and category theory rather than a first topology course.
A short introduction to categorical structures, universal properties and adjunctions.
Lecture notes emphasizing simplicial homology and its technical foundations, followed by singular and cellular methods and the universal coefficient theorem. The recorded version does not develop the cohomology ring.重点讲单纯同调及其技术细节，随后讨论奇异同调、胞腔方法和万有系数定理。这个记录版本没有继续展开上同调环。
Develops infinity-categories, limits, sheaves and infinity-topoi.
Full-text source checked: 2026-11-03
Found via — References: Higher Topos Theory
Develops stable infinity-categories, operads and algebra objects.
Full-text source checked: 2026-11-03
Found via — References: Higher Algebra
A systematic route from homotopical algebra to higher categories.
An online reference for categorical and homotopical foundations, organized by stable tags.
A compact treatment of fundamental algebraic-topological constructions.
Full-text source checked: 2026-11-03
Found via — References: A Concise Course in Algebraic Topology
Develops operadic approaches to iterated loop spaces.
A long introduction spanning low-dimensional groups and higher algebraic K-theory.
Studies complex cobordism and computational tools in stable homotopy.
A recent course relating geometric ideas and higher categorical structures.
Develops univalent foundations through dependent type theory and homotopical ideas.
Uses generalized operads to organize several higher-categorical structures.
Studies surgery-theoretic and topological manifold results. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.
Available portions cover vector bundles, characteristic classes and complex Bott periodicity. The author says only about half the planned book is posted.
Full-text source checked: 2026-11-03
Found via — References: Vector Bundles and K-Theory
Introduces prime and torus decompositions and special classes of three-manifolds. This is an unfinished initial portion, not a complete textbook.
A classical topology course connecting geometric constructions with algebraic invariants. Compare conventions with a contemporary homology course.
Builds symmetric spectra, smash products and stable model structures. This is the complete available unfinished project, not a published edition.
Introduces presentations, quasi-isometries and hyperbolic groups, with fundamental groups and hyperbolic geometry supplying examples.
Develops topological K-theory and the Atiyah–Singer theorem, with equivariant extensions and appendices on Bott periodicity and spectral sequences.
Seminar chapters on vector bundles and their characteristic classes, connecting topological definitions with geometric constructions.讨论向量丛及示性类的讨论班讲义，将拓扑定义与几何构造联系起来。
Develops locally trivial fibrations, vector and principal bundles, classifying spaces and characteristic classes, with connections and curvature.从局部平凡纤维化、向量丛和主丛讲到分类空间与示性类，也讨论联络和曲率。
Found via — References: The Topology of Fiber Bundles
Stable commutator length connects quasimorphisms, bounded cohomology and geometric group theory.
Develops equivariant homotopy theories simultaneously across compact Lie groups using orthogonal spaces and spectra.
A finite-group course on orthogonal spectra, transfers, fixed points and norms; the author labels the notes incomplete.
Connects Dehn functions and nonpositive curvature to recognition and isomorphism problems. The complete lecture manuscript is informal rather than a polished journal treatment.
Relates covering-space L²-invariants to three-manifold topology, linking Betti numbers and torsion with geometric structure.
A systematic treatment of betweenness, convexity and walls, extending to median metric spaces, cube complexes and coarse median structures.
Constructs equivariant spectra and cohomology theories for proper Lie-group actions, with compact isotropy and applications to infinite discrete groups.
代数拓扑基础 — 从前有一只呜喵 (notes credited in the introduction). Work through simplicial chains and relative homology before using the exact-sequence arguments.先计算单纯链与相对同调，再使用后面的正合序列论证。