| WEEK # | TOPICS | READINGS |
|---|---|---|
| 1 | CW-complexes, delta-complexes, simplicial homology, exact sequences, diagram chasing | Chapter 0, pp. 5-8 (PDF)# Chapter 2, Section 2.1, pp. 102-108, exact sequences from ~113 (PDF)# |
| 2 | Singular homology, homotopies and chain homotopies, categories and functors, Eilenberg-Steenrod axioms | Chapter 2, Section 2.1, pp. 104-113 (PDF)# Chapter 2, Section 2.3 (PDF)# |
| 3 | Excision, computations for spheres, equivalence of simplicial and singular homology | Chapter 2, Section 2.3 (PDF)# Chapter 2, Section 2.1, pp. 113-134 (PDF)# |
| 4 | Cellular homology, Mayer-Vietoris sequences, the Mayer-Vietoris argument, homology with coefficients | Chapter 2, Section 2.2, pp. 137-160 (PDF)# |
| 5 | Tensor products, Tor, universal coefficient theorem for homology, products of simplices | Chapter 3, Section 3.A (PDF)# Chapter 3, Section 3.B (PDF)# |
| 6 | The Eilenberg-Zilber shuffle "product" map, diagonal approximations, the Alexander-Whitney map, method of acyclic models, Kunneth formula | No readings in Hatcher text |
| 7 | Duality, cohomology, Ext, universal coefficients for cohomology | Chapter 3, Section 3.1, pp. 190-206 (PDF)# |
| 8 | Projective spaces and Grassmannians, cup products, relative cup products | Chapter 3, Section 3.2, pp. 206-212 (PDF)# |
| 9 | Dual Kunneth formula, field coefficients, cup products in cohomology of projective spaces | Chapter 3, Section 3.2, pp. 212-224 (PDF)# |
| 10 | Manifolds, local orientations, global orientations | Chapter 3, Section 3.3, pp. 230-239 (PDF)# |
| 11 | Cap products and choices of appropriate sign conventions, statement of Poincare duality, limits | Chapter 3, Section 3.3, pp. 239-249 (PDF)# |
| 12 | Compactly supported cohomology, proof of Poincare duality | Chapter 3, Section 3.3, pp. 239-249 (PDF)# |
| 13 | Finish proof of Poincare duality Intersection pairing and cup product | Chapter 3, Section 3.3, pp. 249-252 (PDF)# |
| 14 | Lefschetz fixed point theorem | No readings in Hatcher text |
| 15 | Finish proof of Lefschetz theorem Assorted further topics | No readings in Hatcher text |
Table of Contents
Study Materials
Readings
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The readings are assigned in the textbook for this course:
Hatcher, Allen. Algebraic Topology. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521795401.
The text is freely available online, but paperback copies are also available. We will be focusing on chapters 2 and 3.