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 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 (75%): 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
- In-class participation (25%): This includes
attendance, contributing to class discussion, and occasionally
(once or twice over the course of the semester) presenting
material to the rest of the class.
- HW 1 (Due Wed Sept 10). Hatcher Section 3.1: #1,
6, 11; Section 3.2: #1, 3.
- HW 2 (Due Wed Sept 24). Hatcher Section 3.2: #7,
11, 13; Section 3.3: #1, 8.
- HW 3 (Due Wed Oct 8). Full assignment.
- HW 4 (Due Wed Oct 22). Full assignment.
- HW 5 (Due Wed Nov 5). Hatcher Section 4.2: #1, 12,
15, 28, 30, 34.
- HW 6 (Due Wed Nov 19). Full assignment.
- HW 7 (Due Wed Dec 10). Full assignment.
Here scans of my lecture notes, in PDF format.
- Aug 25. Introduction. Covered pages 1-4.
- Aug 27. Cohomology: examples and
properties. Covered pages 1-4 and 5 through the statement of the
- Aug 29. The Universal Coefficient
Theorem. Did pages 1-5.
- Sept 3. The UCT part II; Intro to the cup product
on cohomology. Did pages 1-5 and also 6 very quickly.
- Sept 5. Definition of the cup product,
basic examples, some properties. Tom's version.
- Sept 8. More on the cup product.
Did pages 1-3 plus the homological example at the top of page 4.
- Sept 10. Cohomology of product
spaces. Covered everything.
- 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.
- Sept 15. Application of cohomology to
division algebras. Covered everything.
- Sept 17. Homology of manifolds. Covered everything.
- Sept 19. Orientations and
homology. Pages 1-4 excepting the outline of the proof of the
- Sept 22. Orientations and
homology: proof of the key lemma. All except uniqueness part of
(b) in step 4 on page 5.
- Sept 24. The statement of Poincaré
Duality and the cap product. Everything but the middle block
on page 4.
- Sept 26. Dual cell structure to a
triangulation. Covered everything.
- Sept 29. First proof of Poincaré
Duality. Through first paragraph on page 4.
- Oct 1. Consequences of duality;
cohomology with compact support. Through top of page 5.
- Oct 3. Direct limits and the general
duality isomorphism. Covered everything.
- Oct 6. Second proof of Poincaré
Duality. Did pages 1-4.
- Oct 8. Other forms of duality.
- Oct 10. Higher homotopy groups.
- Oct 13. Relative homotopy groups.
Through the top section of page 5.
- Oct 15. More on relative homotopy
groups. Did 1-4, though page 4 was rushed.
- Oct 17. Whitehead's Theorem.
- Oct 20. Applications of excision.
- Oct 22. Proof of excision. Did
through middle of 4.
- Oct 24. Rest of proof of excision; intro
to Eilenberg-MacLane spaces. Did 1-top of 5.
- Oct 27. More on Eilenberg-MacLane
spaces. Everything except the proof of the lemma.
- Oct 29. Homology and homotopy: the
Hurewicz homomorphism. Did everything.
- Oct 31. Fibrations and fiber
bundles. Did everything except the proof of local triviality for
the Hopf bundle.
- Nov 3. Fiber bundles. Did pages 1-4.
- Nov 5. Stable homotopy groups. Did
- Nov 7. Cohomology via K(G,
n)’s. Did 1-4.
- Nov 10. Loopspaces and
Ω-spectra. Did 1-4.
- Nov 12. Ω-spectra give cohomology
theories. Did everything.
- Nov 14. No class.
- Nov 17. Representability of
cohomology. Did through top of page 4. The argument in the
middle of page 3 is incomplete, see next lecture.
- Nov 19. Fiber bundles with structure
group. Did 1-middle of 4.
- Nov 21. More on fiber bundles Did
1-3 with a bit of 5.
- Dec 1. Homology with local
coefficients. Did through very top of 5.
- Dec 3. Bordism and homology. Did
- Dec 5. No class.
- Dec 8. Framed bordism and the
Pontryagin-Thom construction. Did 1-4.
- Dec 10. Stably framed bordism and the
stable homotopy groups of spheres.