Mathematics 612, Spring 2025
Algebraic Topology II
Wednesdays, Fridays 1:25-2:40, Physics 259
Office hours: TBA and
by appointment.
Please note that our first class meeting is on Friday January 10.
Classes on Wednesday January 8 follow the Monday schedule.
Course information
Course 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 take-home final exam
for this course.
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, Leray-Serre
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: