Mathematics 262, Fall 2011
Algebraic Topology II
Tuesdays, Thursdays 11:40 am  12:55 pm, Physics 227
Office hours: Tuesdays 1:302:30 and Wednesdays 34
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 takehome 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.

Singular cohomology, cup product, Poincaré duality

Differential forms, de Rham cohomology, Poincaré duality
(again), Künneth Theorem

Vector bundles, Thom isomorphism

Cech cohomology, presheaves

Spectral sequences, double complexes, equivalence of cohomology theories, LeraySerre
spectral sequence

Homotopy groups of spheres, long exact sequence for fiber bundles

Characteristic classes of vector bundles