Mathematics 612, Fall 2014
Algebraic Topology II
Tuesdays, Thursdays 3:054:20pm, Physics 227
Office hours: Mondays 3:004:00, Wednesdays 2:304:00, and by appointment.
Homework assignments

HW 1, due Thursday 9/11:

HW 2, due Thursday 9/25:

HW 3, due Thursday 10/9:

HW 4, due Thursday 10/23:

HW 5, due Thursday 11/6:

HW 6, due Thursday 11/20:

HW 7, due Tuesday 12/2:

Final exam solutions
Course lecture notes
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 takehome 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, LeraySerre
spectral sequence.