Readings

The readings are assigned in the textbook for this course:

Amazon logo 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.

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 (This resource may not render correctly in a screen reader.PDF)

2 Singular homology, homotopies and chain homotopies, categories and functors, Eilenberg-Steenrod axioms

Chapter 2, Section 2.1, pp. 104-113 (This resource may not render correctly in a screen reader.PDF)

Chapter 2, Section 2.3 (This resource may not render correctly in a screen reader.PDF)

3 Excision, computations for spheres, equivalence of simplicial and singular homology

Chapter 2, Section 2.3 (This resource may not render correctly in a screen reader.PDF)

Chapter 2, Section 2.1, pp. 113-134 (This resource may not render correctly in a screen reader.PDF)

4 Cellular homology, Mayer-Vietoris sequences, the Mayer-Vietoris argument, homology with coefficients Chapter 2, Section 2.2, pp. 137-160 (This resource may not render correctly in a screen reader.PDF)
5 Tensor products, Tor, universal coefficient theorem for homology, products of simplices

Chapter 3, Section 3.A (This resource may not render correctly in a screen reader.PDF)

Chapter 3, Section 3.B (This resource may not render correctly in a screen reader.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 (This resource may not render correctly in a screen reader.PDF)
8 Projective spaces and Grassmannians, cup products, relative cup products Chapter 3, Section 3.2, pp. 206-212 (This resource may not render correctly in a screen reader.PDF)
9 Dual Kunneth formula, field coefficients, cup products in cohomology of projective spaces Chapter 3, Section 3.2, pp. 212-224 (This resource may not render correctly in a screen reader.PDF)
10 Manifolds, local orientations, global orientations Chapter 3, Section 3.3, pp. 230-239 (This resource may not render correctly in a screen reader.PDF)
11 Cap products and choices of appropriate sign conventions, statement of Poincare duality, limits Chapter 3, Section 3.3, pp. 239-249 (This resource may not render correctly in a screen reader.PDF)
12 Compactly supported cohomology, proof of Poincare duality Chapter 3, Section 3.3, pp. 239-249 (This resource may not render correctly in a screen reader.PDF)
13

Finish proof of Poincare duality

Intersection pairing and cup product

Chapter 3, Section 3.3, pp. 249-252 (This resource may not render correctly in a screen reader.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