Mathematics 612, Spring 2019
Algebraic Topology II
Tuesdays, Thursdays 1:252:40pm, Physics 227
Office hours: Mondays 2:003:00, Thursdays 2:453:45 (in
other words, right after class), and
by appointment.
Homework assignments

HW 1, due Tuesday 1/22;
solutions to HW 1

HW 2, due Tuesday 1/29;
solutions to HW 2

HW 3, due Tuesday 2/5;
solutions to HW 3

HW 4, due Tuesday 2/12;
solutions to HW 4

HW 5, due Tuesday 2/19;
solutions to HW 5

HW 6, due Tuesday 2/26;
solutions to HW 6

HW 7, due Tuesday 3/5;
solutions to HW 7

HW 8, due Tuesday 3/26;
solutions to HW 8

HW 9, due Tuesday 4/2;
solutions to HW 9

HW 10, due Tuesday 4/9;
solutions to HW 10

HW 11, due Tuesday 4/16;
solutions to HW 11

HW 12, due Tuesday 4/23;
solutions to HW 12
Course lecture notes
For your convenience, my lecture notes from the last time I taught this
course (Fall 2014) are available. I will probably be following these
notes fairly closely though not exclusively. Here they are:
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 but is not required.
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, Poincar?duality
 Differential forms, de Rham cohomology, Poincar?duality
(again), K?neth Theorem
 Vector bundles, Thom isomorphism
 Cech cohomology, presheaves
 Spectral sequences, double complexes, equivalence of cohomology
theories, LeraySerre
spectral sequence.