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. 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.
Your course grade will be based on:
- Homework assignments (85%): 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.
You can check your scores
- In-class participation (15%): This includes attendance and
contributing to class discussion.
- HW 1 (Due Wed Sept 8). Hatcher Section 3.1: #1,
6, 11; Section 3.2: #1, 3.
- HW 2 (Due Wed Sept 22). Hatcher Section 3.2: #7,
11, 13; Section 3.3: #1, 8.
- HW 3 (Due Wed Oct 6). Full
- HW 4 (Due Wed Oct 20). Full
- HW 5 (Due Wed Nov 3). Full
Here are scans of my lecture notes:
- Aug 23. Introduction. Covered
- Aug 25. Cohomology:
examples and properties. Covered pages 1-4 and 5 through the
statement of the general UTC.
- Aug 27. The Universal
Coefficient Theorem. Did pages 1-4.
- Aug 30. The UCT part
II. Did pages 1-4.
- Sept 1. The cup product. Did pages
- Sept 3. More on the cup
product. Did pages 1-3.
- Sept 8. Cohomology of
products. Covered everything.
- Sept 10. Cohomology of
products II. Covered everything.
- Sept 13. Division algebras; intro to
manifolds. 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 1-4.
- Sept 27. First proof of Poincaré
Duality. Covered 1-3 except for the proof at the bottom of
- Sept 29. Consequences of duality;
cohomology with compact support. Through page 3.
- Oct 1. Direct limits and the general
duality isomorphism. Through top half of page 4.
- Oct 4. Second proof of Poincaré
Duality. Did pages 1-3.
- Oct 6. Other forms of
duality. Covered 1-4.
- Oct 8. A little more duality.
- Oct 11. Higher homotopy
groups. Also did homotopy groups of a product.
- Oct 13. Relative homotopy groups.
Through the first proof on page 5.
- Oct 15. More on
relative homotopy groups. Did 1-4.
- Oct 18. Whitehead's
Theorem. Started midway down page 2. Covered everything.
- Oct 20. Applications of excision.
- 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.
- Oct 25. 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.
- Nov 1. 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 8. 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 15. Representability of
cohomology. Did through top of page 4. The argument in the
middle of page 3 is incomplete, see next lecture.
- Nov 17. Fiber bundles with structure
group. Did 1-middle of 4.
- Nov 19. More on fiber bundles Did
1-3 with a bit of 5.
- Nov 29. Homology with local
coefficients. Did through very top of 5.
- Dect 1. Bordism and homology. Did
- Dec 3. Framed bordism and the
Pontryagin-Thom construction. Did 1-4.
- Dec 6. Stably framed bordism and the
stable homotopy groups of spheres.
- Dec 8. TBA/slack.