Columbia University G4307
Algebraic topology I

Basic information

Room/Time: MW 1:10pm-2:25pm, 507 Math
Instructor: Mikhail Khovanov
Office: 620 Math
E-mail:
Final exam: TBA
Webpage: www.math.columbia.edu/~khovanov/gradat2014
 

Textbooks

We will use Algebraic Topology by Alan Hatcher as our primary textbook. It is free to download and the printed version is inexpensive.

An additional and excellent textbook is 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)
Homotopy groups, cellular approximations, fibrations, Eilenberg-MacLane spaces. (Fuchs-Fomenko-Gutenmacher)
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)
If time allows: Steenrod squares.

Homework

Homework will be assigned on Mondays, due Monday 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 15.      
Homework 2, due September 22.      
Homework 3, due September 29.      
Homework 4, due October 13.      
Homework 5, due October 20.      
Homework 6, due October 27.      
Homework 7, due November 17.      
Homework 8, due November 24.      
Homework 9, due December 1.      

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.