Mathematics 612, Fall 2014

Algebraic Topology II

Tuesdays, Thursdays 3:05-4:20pm, Physics 227
Office hours: Mondays 3:00-4:00, Wednesdays 2:30-4:00, and by appointment.

Homework assignments

HW 1, due Thursday 9/11:
- HW 1 part 1, assigned 8/28; LaTeX source
- HW 1 part 2, assigned 9/4; LaTeX source
- HW 1 solutions
HW 2, due Thursday 9/25:
- HW 2 part 1, assigned 9/11; LaTeX source
- HW 2 part 2, assigned 9/18; LaTeX source
- HW 2 solutions
HW 3, due Thursday 10/9:
- ~~HW 3 part 1, assigned 9/25; LaTeX source~~
- HW 3 (complete, parts 1 and 2), assigned 10/2; LaTeX source
- HW 3 solutions
HW 4, due Thursday 10/23:
- HW 4 part 1, assigned 10/9; LaTeX source
- HW 4 part 2, assigned 10/16; LaTeX source
- HW 4 solutions
HW 5, due Thursday 11/6:
- HW 5 part 1, assigned 10/23; LaTeX source
- HW 5 part 2, assigned 10/30; LaTeX source
- HW 5 solutions
HW 6, due Thursday 11/20:
- HW 6 part 1, assigned 11/6; LaTeX source
- HW 6 part 2, assigned 11/13; LaTeX source
- HW 6 solutions
HW 7, due Tuesday 12/2:
- HW 7, assigned 11/20; LaTeX source
- HW 7 solutions
Final exam solutions

Course lecture notes

Part 1: simplicial and singular cohomology
Part 2: de Rham cohomology
Part 3: Cech cohomology
Part 4: spectral sequences
Part 5: Thom isomorphism

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.