Columbia University G4307
Algebraic topology I
Basic information
Call number: | 10014
|
Room/Time: | MW 2:40pm-3:55pm, 507 Math |
Discussions: | Thursday 4-5:30pm in 528 Math |
TA: | Krzysztof Putyra, putyra@math.columbia.edu |
TA office hours: | Wednesday 4:15-5:45pm in 408 Math
|
Instructor: | Mikhail Khovanov |
Office: | 620 Math |
Office Hours: | Monday 1:30-2:30pm or you can stop by any
time |
E-mail: |
khovanov@math.columbia.edu |
Final exam: | TBA | |
Webpage: | www.math.columbia.edu/~khovanov/gradalgtop | |
|
Textbooks
We will use
Algebraic
Topology by Alan Hatcher as our primary textbook.
It is free to download and the printed version is inexpensive.
It can be nicely supplemented by
Homotopic topology by A.Fomenko, D.Fuchs, and V.Gutenmacher.
The first two chapters cover the material of the fall semester.
Chapters 1 and 2: Homotopy and Homology,
Chapter 3: Spectral sequences,
Chapter 4: Cohomology operations,
Chapter 5: The Adams spectral sequence,
Index.
Syllabus
CW complexes and cofibrations. (Hatcher, Chapter 0)
Fundamental group and covering spaces. (Hatcher, Chapter 1)
Homology. Singular and simplicial homology, Mayer-Vietoris sequences,
coefficients. (Hatcher, Chapter 2)
Cohomology, universal coefficient theorem. Products in homology and
cohomology. Kunneth formula. Poincare duality. (Hatcher, Chapter 3)
Homotopy groups, cellular approximation. Fibrations and Serre
fibrations. Weak equivalence. Computations. Eilenberg-MacLane
spaces. (Hatcher, Chapter 4)
If time allows: cohomology of groups, local coefficients, computations
with Ext groups, derived functors.
Homework
Homework will be assigned on Wednesdays, due Wednesday the following week
before class. Homework and the final exam will contribute 70% and
30%, respectively, to the overall grade. The lowest (normalized)
homework score will be dropped.
Homework 1, due September 11.
Homework 2, due September 18.
Homework 3, due September 25.
Homework 4, due October 2.
Homework 5, due October 9.
Homework 6, due October 16.
Homework 7, due October 23.
Homework 8, due October 30.
Homework 9, due November 13.
Homework 10, due November 20.
Homework 11, due November 27.
Additional resources
Online books
Boris Botvinnik
Lecture Notes on
Algebraic Topology.
James F. Davis and Paul Kirk
Lecture Notes in Algebraic Topology.
Peter May
Concise Course in Algebraic Topology.
Online Course Materials
Algebraic Topology II by Mark Behrens.
Homotopy theory by Martin Frankland.
Homotopy theory course by Bert Guillou.