Mathematics 262, Fall 2011
Algebraic Topology II
Tuesdays, Thursdays 11:40 am - 12:55 pm, Physics 227
Office hours: Tuesdays 1:30-2:30 and Wednesdays 3-4
Homework assignments
- HW 1, due September 8: PDF
| TeX source | Solutions
- HW 2, due September 15: PDF
| TeX source | Solutions
- HW 3, due September 22: PDF
| TeX source | Solutions
- HW 4, due September 29: PDF | TeX source | Solutions
- HW 5 (corrected), due October 6: PDF | TeX source | Solutions
- HW 6, due October 20: PDF
| TeX source
| Solutions
- HW 7, due October 27: PDF
| TeX source |
Solutions
- HW 8, due November 3: PDF
| TeX source |
Solutions
- HW 9, due November 10: PDF
| TeX source |
Solutions
- HW 10, due November 17: PDF
| TeX source |
Solutions
- HW 11 (final version), due December 1: PDF
| TeX source |
Solutions
Course information
Textbook: Differential Forms in Algebraic Topology,
by R. Bott and L. Tu. Another text that will be useful
for the first portion of the course, when we discuss singular
cohomology, is Algebraic Topology by A. Hatcher. I also
recommend Characteristic Classes by J. Milnor and J. Stasheff
as a great reference, but we won't cover much of the material from
it.
Prerequisites: Math 261 or familiarity with equivalent material
(fundamental group, simplicial/singular homology, CW complexes;
essentially the first two chapters of Hatcher). Math 267 or familiarity
with basic differential topology (smooth manifolds, tangent/cotangent
bundle, differential forms) will also be helpful.
There
will be weekly homework assignments and a take-home final exam for this
course.
Here are topics that I plan to cover in the course. Topics near the
end of this list (principally the last two bullet points) are probably
wishful thinking, but we should be able to cover the rest.
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Singular cohomology, cup product, Poincaré duality
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Differential forms, de Rham cohomology, Poincaré duality
(again), Künneth Theorem
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Vector bundles, Thom isomorphism
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Cech cohomology, presheaves
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Spectral sequences, double complexes, equivalence of cohomology theories, Leray-Serre
spectral sequence
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Homotopy groups of spheres, long exact sequence for fiber bundles
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Characteristic classes of vector bundles