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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.
Readings files.
| 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 |