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 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. Three 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.

Grading

Your course grade will be based on:

Homework Assignments

Lecture notes

Here are scans of my lecture notes:

  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 1-4.
  4. Aug 30. The UCT part II. Did pages 1-4.
  5. Sept 1. The cup product. Did pages 1-5.
  6. Sept 3. More on the cup product. Did pages 1-3.
  7. Sept 8. Cohomology of products. Covered everything.
  8. Sept 10. Cohomology of products II. Covered everything.
  9. Sept 13. Division algebras; intro to manifolds. Covered everything.
  10. Sept 15. Homology of manifolds. Covered everything.
  11. Sept 17 Orientations and homology. Pages 1-4 excepting the outline of the proof of the lemma.
  12. Sept 20. Orientations and homology: proof of the key lemma. All except uniqueness part of (b) in step 4 on page 5.
  13. Sept 22. The statement of Poincaré Duality and the cap product. Everything but the middle block on page 4.
  14. Sept 24. Dual cell structure to a triangulation. Covered 1-4.
  15. Sept 27. First proof of Poincaré Duality. Covered 1-3 except for the proof at the bottom of 3.
  16. Sept 29. Consequences of duality; cohomology with compact support. Through page 3.
  17. Oct 1. Direct limits and the general duality isomorphism. Through top half of page 4.
  18. Oct 4. Second proof of Poincaré Duality. Did pages 1-3.
  19. Oct 6. Other forms of duality. Covered 1-4.
  20. Oct 8. A little more duality.
  21. Oct 11. Higher homotopy groups. Also did homotopy groups of a product.
  22. Oct 13. Relative homotopy groups. Through the first proof on page 5.
  23. Oct 15. More on relative homotopy groups. Did 1-4.
  24. Oct 18. Whitehead's Theorem. Started midway down page 2. Covered everything.
  25. Oct 20. Applications of excision. Did 1-4.
  26. Oct 22. Proof of excision. Did through middle of 4.

Below are my lecture notes from the last time I taught this course (2014), offered as a preview of forthcoming attractions.

  1. Oct 25. Rest of proof of excision; intro to Eilenberg-MacLane spaces. Did 1-top of 5.
  2. Oct 27. More on Eilenberg-MacLane spaces. Everything except the proof of the lemma.
  3. Oct 29. Homology and homotopy: the Hurewicz homomorphism. Did everything.
  4. Nov 1. Fibrations and fiber bundles. Did everything except the proof of local triviality for the Hopf bundle.
  5. Nov 3. Fiber bundles. Did pages 1-4.
  6. Nov 5. Stable homotopy groups. Did everything.
  7. Nov 8. Cohomology via K(G, n)’s. Did 1-4.
  8. Nov 10. Loopspaces and Ω-spectra. Did 1-4.
  9. Nov 12. Ω-spectra give cohomology theories. Did everything.
  10. Nov 15. Representability of cohomology. Did through top of page 4. The argument in the middle of page 3 is incomplete, see next lecture.
  11. Nov 17. Fiber bundles with structure group. Did 1-middle of 4.
  12. Nov 19. More on fiber bundles Did 1-3 with a bit of 5.
  13. Nov 29. Homology with local coefficients. Did through very top of 5.
  14. Dect 1. Bordism and homology. Did 1-4.
  15. Dec 3. Framed bordism and the Pontryagin-Thom construction. Did 1-4.
  16. Dec 6. Stably framed bordism and the stable homotopy groups of spheres.
  17. Dec 8. TBA/slack.

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