Columbia University W4053
Introduction to algebraic topology

Basic information

Call number: 10123
Room/Time: Tu, Th 2:40pm-3:55pm, 407 Math
Discussions: 5:30-7pm in room 622
TA: Krzysztof Putyra,
TA office hours: Monday 1-4pm in Math 406 (Help room)
Instructor: Mikhail Khovanov
Office: 620 Math
Office Hours: Tuesday 4-5pm, Thursday 10:30-11:30am or by appointment
Midterm: Thursday, March 14
Final exam: Take-home


We will be using the following two textbooks:

1) Homotopic topology, by A.Fomenko, D.Fuchs, and V.Gutenmacher.
Chapters 1 and 2: Homotopy and Homology,
Chapter 3: Spectral sequences,
Chapter 4: Cohomology operations,
Chapter 5: The Adams spectral sequence,

2) Algebraic Topology by Alan Hatcher, Cambridge U Press. Free download; printed version can be bought cheaply online.


Homotopy theory:

Homotopy and homotopy equivalence. CW complexes. Cellular approximation.
Category theory, functors and adjointness.
Fundamental group and its computation. Coverings and their classification.
Fibrations and Serre fibrations. Relative homotopy groups.
Complexes and exact sequences. Homotopy sequence of a fibration.
Homotopy groups of CW complexes.
Weak equivalence and cellular approximation. Eilenberg-Maclane spaces.

Homology theory:

Chain complexes and chain maps. Homology of complexes.
Singular homology, homology of CW comlexes, computations.
Homotopy and homology, Hurewicz theorem.
Cohomology groups. Homology and cohomology with coefficients.
Kunneth formula. Multiplications in cohomology.
Applications of homology and cohomology.
Manifolds, Poincare duality.



Homework will be assigned on Tuesdays, due Tuesday the following week before class. The numerical grade for the course will be the following linear combination: 50% homework, 5% quizzes, 15% midterm, 30% final. The lowest homework score will be dropped.
Homework 1      
Homework 2      
Homework 3      
Homework 4      
Homework 5      
Homework 6      
Homework 7      
Homework 8      
Homework 9      
Homework 10      
Homework 11      
Homework 12      

Quiz 1
Quiz 2
Final exam

Additional online resources

Lecture notes on algebraic topology by David Wilkins.
Homotopy theory course by Bert Guillou.
Algebraic Topology II by Mark Behrens.
Peter May's Concise Course in Algebraic Topology.