Mathematics 612, Spring 2024
Algebraic Topology II
Mondays, Wednesdays, 1:252:40, Physics 259
Office hours: Wednesdays 4:005:00, Thursdays 12:001:00, and
by appointment.
Course information
Course syllabus (updated 1/24)  all of the
information below is also contained in the syllabus.
My plan is to post homework assignments and solutions on Canvas.
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 (the first few weeks), when we
discuss singular cohomology. The rest of the course will be based on
Bott and Tu, which is the "official" text for the course.
Prerequisite: Math 611 or familiarity with equivalent material
(fundamental group, simplicial/singular homology, CW complexes;
essentially the first two chapters of Hatcher). Math 620 or familiarity
with basic differential topology (smooth manifolds, tangent/cotangent
bundle, differential forms) will also be assumed, but this isn't an
ironclad prerequisite; please talk to me if you don't have previous
background in smooth manifolds.
Grading: There
will be weekly homework assignments and a takehome final exam
for this course.
Course meetings: I will be out of town on February 12 and March 25.
There will be makeup classes tentatively scheduled for February 16 and
March 29. Please let me know if you have issues with this scheduling.
Here are the topics that I plan to cover in the course:

Singular cohomology, cup product, Poincaré duality
 Differential forms, de Rham cohomology, Poincaré duality
(again but now via de Rham cohomology), Künneth Theorem
 Čech cohomology, presheaves
 Spectral sequences, double complexes, equivalence of cohomology
theories, LeraySerre
spectral sequence
 (to the extent that time permits:) Vector bundles, Thom isomorphism.
Course lecture notes
For your convenience, my lecture notes from a previous time I taught this
course (Fall 2014) are available. I will probably be following these
notes fairly closely though not exclusively. Here they are: