Mathematics 612, Fall 2013

Algebraic Topology II

Wednesdays, Fridays 1:25-2:40pm, Physics 227
Office hours: Wednesdays 4:20-5:00, Thursdays 12:00-2:00, or drop by or make an appointment. (Please note that my undergraduate course has priority for the official office hours.)

There will be no class November 6 and November 8.

Homework assignments

HW 1 part 1 and HW 1 part 2, due Friday 9/13; solutions to HW 1
HW 2 part 1 and HW 2 part 2, due Friday 9/27; solutions to HW 2
HW 3 part 1 (LaTeX source) and HW 3 part 2 (LaTeX source), due Friday 10/11; solutions to HW 3
HW 4 part 1 (LaTeX source) and HW 4 part 2 (LaTeX source), due Friday 10/25; solutions to HW 4
HW 5 (LaTeX source), due Friday 11/15; solutions to HW 5
HW 6 part 1 (LaTeX source) and HW 6 part 2 (LaTeX source), due Wednesday 12/4; solutions to HW 6

Course information

Textbooks:

Algebraic Topology by Allen Hatcher
Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu.

We will use Hatcher for the first portion of the course, when we discuss singular cohomology. The rest of the course will be based on Bott and Tu.

Prerequisites: Math 611 or familiarity with equivalent material (fundamental group, simplicial/singular homology, CW complexes; essentially the first two chapters of Hatcher). Math 621 or familiarity with basic differential topology (smooth manifolds, tangent/cotangent bundle, differential forms) will also be helpful.

There will be weekly/biweekly homework assignments and a take-home final exam for this course.

Here are topics that I plan to cover in the course:

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, Leray-Serre spectral sequence.