# Math 526, Algebraic Topology II

## Fall 2021

### Course Description

Math 526 is a second course in algebraic topology. It develops the theory of cohomology, which is homology's algebraically dual sibling, and applies it to a wide range of geometric problems. A key advantage of cohomology over homology is that it has a multiplication, called the cup product, which makes it into a ring; for manifolds, this product corresponds to the exterior multiplication of differential forms. The course includes the study of PoincarĂ© duality which interrelates the (co)homology of a given manifold in different dimensions, as well as topics such as the Kunneth formula and the universal coefficient theorem.

The other major topic covered in this course are the higher homotopy groups, including things like cellular approximation, Whitehead's theorem, excision, the Hurewicz Theorem, Eilenberg-MacLane spaces, and representability of cohomology.

Basically, we'll cover Chapters 3-4 of the required text, which is

• Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. ISBN: 0521795400

You can download the full text for free here. A useful list of errata is also available. Two other helpful perspectives on this material, which are quite different from both Hatcher and each other, are:

Additional topics covered will depend on audience interest, but may include spectral sequences, basic sheaf cohomology, and homology with local coefficients. Sources here include

Prerequisites: Math 525 and Math 500 or similar.

• Homework assignments: These will be roughly biweekly (so 7 assignments in total), and typically due on Wednesdays. Late homework will not be accepted; however, your lowest two homework grades will be dropped so you are effectively allowed two infinitely late assignments. Collaboration on homework is permitted, nay encouraged. However, you must write up your solutions individually.
• In-class participation: This includes attendance and contributing to class discussion.

• TBA.

### Lecture notes

Here are scans of my lecture notes from the last time I taught this course (2014), I will update them as we go along.

1. Aug 23. Introduction. Covered pages 1-4.
2. Aug 25. Cohomology: examples and properties. Covered pages 1-4 and 5 through the statement of the general UTC.
3. Aug 27. The Universal Coefficient Theorem. Did pages 1-5.
4. Aug 30. The UCT part II; Intro to the cup product on cohomology. Did pages 1-5 and also 6 very quickly.
5. Sept 1. Definition of the cup product, basic examples, some properties. Tom's version.
6. Sept 3. More on the cup product. Did pages 1-3 plus the homological example at the top of page 4.
7. Sept 6. Labor Day.
8. Sept 8. Cohomology of product spaces. Covered everything.
9. Sept 10. Cohomology of product spaces II. Did pages 1-(middle of 5) and 7. There are some errors in these notes, see the correction in the next lecture.
10. Sept 13. Application of cohomology to division algebras. Covered everything.
11. Sept 15. Homology of manifolds. Covered everything.
12. Sept 17 Orientations and homology. Pages 1-4 excepting the outline of the proof of the lemma.
13. Sept 20. Orientations and homology: proof of the key lemma. All except uniqueness part of (b) in step 4 on page 5.
14. Sept 22. The statement of PoincarĂ© Duality and the cap product. Everything but the middle block on page 4.
15. Sept 24. Dual cell structure to a triangulation. Covered everything.
16. Sept 27. First proof of PoincarĂ© Duality. Through first paragraph on page 4.
17. Sept 29. Consequences of duality; cohomology with compact support. Through top of page 5.
18. Oct 1. Direct limits and the general duality isomorphism. Covered everything.
19. Oct 4. Second proof of PoincarĂ© Duality. Did pages 1-4.
20. Oct 6. Other forms of duality. Covered everything.
21. Oct 8. Higher homotopy groups.
22. Oct 11. Relative homotopy groups. Through the top section of page 5.
23. Oct 13. More on relative homotopy groups. Did 1-4, though page 4 was rushed.
25. Oct 18. Applications of excision. Did 1-4.
26. Oct 20. Proof of excision. Did through middle of 4.
27. Oct 22. Rest of proof of excision; intro to Eilenberg-MacLane spaces. Did 1-top of 5.
28. Oct 25. More on Eilenberg-MacLane spaces. Everything except the proof of the lemma.
29. Oct 27. Homology and homotopy: the Hurewicz homomorphism. Did everything.
30. Oct 29. Fibrations and fiber bundles. Did everything except the proof of local triviality for the Hopf bundle.
31. Nov 1. Fiber bundles. Did pages 1-4.
32. Nov 3. Stable homotopy groups. Did everything.
33. Nov 5. Cohomology via K(G, n)’s. Did 1-4.
34. Nov 8. Loopspaces and Ω-spectra. Did 1-4.
35. Nov 10. Ω-spectra give cohomology theories. Did everything.
36. Nov 12. Representability of cohomology. Did through top of page 4. The argument in the middle of page 3 is incomplete, see next lecture.
37. Nov 15. Fiber bundles with structure group. Did 1-middle of 4.
38. Nov 17. More on fiber bundles Did 1-3 with a bit of 5.
39. Nov 19. Homology with local coefficients. Did through very top of 5.
40. Nov 29. Bordism and homology. Did 1-4.
41. Dec 1. Framed bordism and the Pontryagin-Thom construction. Did 1-4.
42. Dec 3. Stably framed bordism and the stable homotopy groups of spheres.
43. Dec 6. TBA/slack.
44. Dec 8. TBA/slack.