Math 526, Algebraic Topology II

Fall 2014

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 Poincaré 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

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.


Your course grade will be based on:

Homework Assignments

Lecture notes

Here scans of my lecture notes, in PDF format.

  1. Aug 25. Introduction. Covered pages 1-4.
  2. Aug 27. Cohomology: examples and properties. Covered pages 1-4 and 5 through the statement of the general UTC.
  3. Aug 29. The Universal Coefficient Theorem. Did pages 1-5.
  4. Sept 3. The UCT part II; Intro to the cup product on cohomology. Did pages 1-5 and also 6 very quickly.
  5. Sept 5. Definition of the cup product, basic examples, some properties. Tom's version.
  6. Sept 8. More on the cup product. Did pages 1-3 plus the homological example at the top of page 4.
  7. Sept 10. Cohomology of product spaces. Covered everything.
  8. Sept 12. 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.
  9. Sept 15. Application of cohomology to division algebras. Covered everything.
  10. Sept 17. Homology of manifolds. Covered everything.
  11. Sept 19. Orientations and homology. Pages 1-4 excepting the outline of the proof of the lemma.
  12. Sept 22. Orientations and homology: proof of the key lemma. All except uniqueness part of (b) in step 4 on page 5.
  13. Sept 24. The statement of Poincaré Duality and the cap product. Everything but the middle block on page 4.
  14. Sept 26. Dual cell structure to a triangulation. Covered everything.
  15. Sept 29. First proof of Poincaré Duality. Through first paragraph on page 4.
  16. Oct 1. Consequences of duality; cohomology with compact support. Through top of page 5.
  17. Oct 3. Direct limits and the general duality isomorphism. Covered everything.
  18. Oct 6. Second proof of Poincaré Duality. Did pages 1-4.
  19. Oct 8. Other forms of duality. Covered everything.
  20. Oct 10. Higher homotopy groups.
  21. Oct 13. Relative homotopy groups. Through the top section of page 5.
  22. Oct 15. More on relative homotopy groups. Did 1-4, though page 4 was rushed.
  23. Oct 17. Whitehead's Theorem.
  24. Oct 20. Applications of excision. Did 1-4.
  25. Oct 22. Proof of excision. Did through middle of 4.
  26. Oct 24. Rest of proof of excision; intro to Eilenberg-MacLane spaces. Did 1-top of 5.
  27. Oct 27. More on Eilenberg-MacLane spaces. Everything except the proof of the lemma.
  28. Oct 29. Homology and homotopy: the Hurewicz homomorphism. Did everything.
  29. Oct 31. Fibrations and fiber bundles. Did everything except the proof of local triviality for the Hopf bundle.
  30. Nov 3. Fiber bundles. Did pages 1-4.
  31. Nov 5. Stable homotopy groups. Did everything.
  32. Nov 7. Cohomology via K(G, n)’s. Did 1-4.
  33. Nov 10. Loopspaces and Ω-spectra. Did 1-4.
  34. Nov 12. Ω-spectra give cohomology theories. Did everything.
  35. Nov 14. No class.
  36. Nov 17. Representability of cohomology. Did through top of page 4. The argument in the middle of page 3 is incomplete, see next lecture.
  37. Nov 19. Fiber bundles with structure group. Did 1-middle of 4.
  38. Nov 21. More on fiber bundles Did 1-3 with a bit of 5.
  39. Dec 1. Homology with local coefficients. Did through very top of 5.
  40. Dec 3. Bordism and homology. Did 1-4.
  41. Dec 5. No class.
  42. Dec 8. Framed bordism and the Pontryagin-Thom construction. Did 1-4.
  43. Dec 10. Stably framed bordism and the stable homotopy groups of spheres.

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