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.