Algebraic Topology I

About the course

This is the first semester of the graduate algebraic topology sequence. The guiding idea of the subject is to attach algebraic invariants — groups, modules, chain complexes — to topological spaces in a functorial way, and to use these invariants to distinguish spaces and to prove theorems inaccessible by point-set methods alone.

Prerequisites. The equivalent of one semester each of graduate abstract algebra and real analysis, including point-set topology. Prior exposure to the fundamental group (e.g., at the level of AS.110.413) is helpful but not assumed. Comfort with modules over a ring and basic categorical language will be assumed after the first two weeks.

Lecture notes

Syllabus

Part I — Categories and homological algebra

• Week 1Categories, functors, natural transformations. Limits and colimits; adjoint functors. Additive categories; kernels, cokernels, and abelian categories.
• Week 2Chain complexes and homology as a functor. Chain homotopy. The long exact sequence of a short exact sequence of complexes; the snake lemma and diagram chases. Projective and injective objects; resolutions.
• Week 3Derived functors: Tor and Ext. Universal coefficient and Künneth theorems (algebraic form). The homotopy category K(𝒜).
• Week 4Localization of categories; the derived category D(𝒜). Triangulated structure: distinguished triangles as the replacement for short exact sequences. Derived functors from this viewpoint.

Part II — Simplicial sets and singular homology

• Week 5The simplex category Δ; simplicial sets, faces and degeneracies. Geometric realization and the singular simplicial set; the realization–singular adjunction. Simplicial abelian groups and the associated chain complex; statement of Dold–Kan.
• Week 6Singular homology: definition, functoriality, H₀ and π₀, homology of a point. Homotopy invariance via simplicial/prism arguments.
• Week 7Relative homology. Excision and Mayer–Vietoris. Degree of self-maps of Sⁿ; Brouwer fixed point theorem; invariance of dimension and domain.
• Week 8CW complexes and cellular homology; comparison with singular homology. Euler characteristic. Homology with coefficients; topological universal coefficients and Künneth (survey); Lefschetz fixed point theorem (time permitting).

Part III — Homotopy theory and the fundamental group

• Week 9Homotopy groups π_n: definitions, basepoints, the action of π₁ on π_n. Statement of Whitehead's theorem and the Hurewicz theorem (proofs deferred to 616).
• Week 10Fibrations: the homotopy lifting property, Serre fibrations, fiber bundles as examples. The long exact sequence of a fibration, with the Hopf fibration S¹ → S³ → S² and the computation of π₃(S²) as the flagship application. Path–loop fibration.
• Week 11The fundamental group in detail: π₁(S¹) via lifting, applications (fundamental theorem of algebra, Borsuk–Ulam in low dimension). Free products and the Seifert–van Kampen theorem.
• Week 12Covering spaces: lifting criteria, the classification of coverings via subgroups of π₁, universal covers, deck transformations, and the Galois correspondence. Covering space proofs of group-theoretic facts (subgroups of free groups are free); preview of 616.

Final project presentations

• Week 13–14In-class final project presentations (Nov 30 – Dec 9). No new lecture material.

Thanksgiving break (late November): no classes.

Grading

Your course grade is based entirely on class participation/attendance and a final project. There are no exams.

• Participation and attendance (10%). This is a lecture course, but a conversational one: attend regularly, ask and answer questions, and occasionally present an argument at the board when invited. The participation grade reflects consistent, good-faith engagement over the semester — not the number of times you speak, and certainly not whether your questions are "good enough." If you must miss more than an occasional class (illness, travel, conflicts), just let me know.
• Final project (90%). An in-class presentation on a topic extending the course, to take place during the last two weeks of class (Nov 30 – Dec 9). There is no written paper. Suitable topics include: the Dold–Kan correspondence in full; model structures on simplicial sets; group cohomology via derived functors; the Hurewicz theorem; classifying spaces and principal bundles; Poincaré duality; the Lefschetz fixed point theorem; covering space theory from the Galois-theoretic viewpoint; braid groups and configuration spaces. Topic proposal due mid-October; optional draft of your presentation materials early November. Graded on mathematical correctness, depth of engagement with the material, and clarity of exposition — not originality of research.
• Homework (assigned, not counted toward the grade). Problem sets roughly every one to two weeks, graded for your benefit only and returned with comments. Collaboration is encouraged; write up solutions in your own words.

Problem sets

• Homework 1 (PDF) — Additive and abelian categories, chain complexes, and chain homotopies. LaTeX source.

Sets are graded for your benefit and returned with comments; they do not count toward the course grade.

Important dates

Policies

• Collaboration and integrity. Discussion of homework with classmates is encouraged. The final project must be your own exposition; all sources (including AI tools, if used) must be cited. All work is subject to the university's graduate academic misconduct policies.
• AI use. Prohibited on homework: the problem sets exist entirely for your benefit, so outsourcing them defeats the purpose. Permitted for everything else, subject to the citation requirement for the final project.
• Devices in class. Discouraged. If your attention drifts, work on something productive instead — a lit screen makes it harder for everyone nearby to pay attention.
• Accommodations. Students requiring accommodations should register with Student Disability Services and contact me early in the semester.

Texts

No single text covers the course in the order we follow. Primary references, all freely or electronically available:

• A. Hatcher, Algebraic Topology — free from the author's page. Main reference for homology, the fundamental group, and covering spaces.
• C. Weibel, An Introduction to Homological Algebra — Cambridge University Press; main reference for Part I.
• S. Gelfand and Y. Manin, Methods of Homological Algebra — SpringerLink; for derived and triangulated categories.
• P. Goerss and J. F. Jardine, Simplicial Homotopy Theory — SpringerLink; main reference for simplicial sets.
• J. P. May, A Concise Course in Algebraic Topology — free PDF from the author's page; secondary reference throughout, especially for fibrations.