Math 526, Algebraic Topology II
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
- Allen Hatcher, Algebraic Topology, Cambridge University
Press, 2002. ISBN: 0521795400
You can download the full text for free
A useful list of
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: 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.
Here are scans of my lecture notes from the last time I taught this
course (2014), I will update them as we go along.
- Aug 23. Introduction. Covered pages 1-4.
- Aug 25. Cohomology: examples and
properties. Covered pages 1-4 and 5 through the statement of the
- Aug 27. The Universal Coefficient
Theorem. Did pages 1-5.
- Aug 30. The UCT part II; Intro to the cup product
on cohomology. Did pages 1-5 and also 6 very quickly.
- Sept 1. Definition of the cup product,
basic examples, some properties. Tom's version.
- Sept 3. More on the cup product.
Did pages 1-3 plus the homological example at the top of page 4.
- Sept 6. Labor Day.
- Sept 8. Cohomology of product
spaces. Covered everything.
- 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.
- Sept 13. Application of cohomology to
division algebras. Covered everything.
- Sept 15. Homology of manifolds. Covered everything.
- Sept 17 Orientations and
homology. Pages 1-4 excepting the outline of the proof of the
- Sept 20. Orientations and
homology: proof of the key lemma. All except uniqueness part of
(b) in step 4 on page 5.
- Sept 22. The statement of Poincaré
Duality and the cap product. Everything but the middle block
on page 4.
- Sept 24. Dual cell structure to a
triangulation. Covered everything.
- Sept 27. First proof of Poincaré
Duality. Through first paragraph on page 4.
- Sept 29. Consequences of duality;
cohomology with compact support. Through top of page 5.
- Oct 1. Direct limits and the general
duality isomorphism. Covered everything.
- Oct 4. Second proof of Poincaré
Duality. Did pages 1-4.
- Oct 6. Other forms of duality.
- Oct 8. Higher homotopy groups.
- Oct 11. Relative homotopy groups.
Through the top section of page 5.
- Oct 13. More on relative homotopy
groups. Did 1-4, though page 4 was rushed.
- Oct 15. Whitehead's Theorem.
- Oct 18. Applications of excision.
- Oct 20. Proof of excision. Did
through middle of 4.
- Oct 22. Rest of proof of excision; intro
to Eilenberg-MacLane spaces. Did 1-top of 5.
- Oct 25. More on Eilenberg-MacLane
spaces. Everything except the proof of the lemma.
- Oct 27. Homology and homotopy: the
Hurewicz homomorphism. Did everything.
- Oct 29. Fibrations and fiber
bundles. Did everything except the proof of local triviality for
the Hopf bundle.
- Nov 1. Fiber bundles. Did pages 1-4.
- Nov 3. Stable homotopy groups. Did
- Nov 5. Cohomology via K(G,
n)’s. Did 1-4.
- Nov 8. Loopspaces and
Ω-spectra. Did 1-4.
- Nov 10. Ω-spectra give cohomology
theories. Did everything.
- Nov 12. Representability of
cohomology. Did through top of page 4. The argument in the
middle of page 3 is incomplete, see next lecture.
- Nov 15. Fiber bundles with structure
group. Did 1-middle of 4.
- Nov 17. More on fiber bundles Did
1-3 with a bit of 5.
- Nov 19. Homology with local
coefficients. Did through very top of 5.
- Nov 29. Bordism and homology. Did
- Dec 1. Framed bordism and the
Pontryagin-Thom construction. Did 1-4.
- Dec 3. Stably framed bordism and the
stable homotopy groups of spheres.
- Dec 6. TBA/slack.
- Dec 8. TBA/slack.