Columbia University GU4053
Introduction to algebraic topology

Basic information

Call number: 12625
Room/Time: TuTh 2:40pm-3:55pm, 417 Math
Graduate Teaching Assistant:    Song Yu,   email:
Graduate TA office hours: Tuesday 11am-2pm in Help Room, 406 Math
Instructor: Mikhail Khovanov
Office: 620 Math
Office Hours: Tentative: Zoom Wednesdays 3-4pm (tentative) and Office (620 Math) Thursdays 1-2pm or by appointment
Midterm: Tuesday, March 7 (tentative)
Final exam: Take-home

Prerequisites: An undergraduate topology course, covering point-set topology and introduction to the fundamental group.


We will be using the following two textbooks:
1) Algebraic Topology by Alan Hatcher, Cambridge U Press. Free download; printed version can be bought cheaply online.

2) A.Fomenko, D.Fuchs, Homotopical topology. Pdf version can be downloaded from the Columbia library website or via Springerlink site for the book. Springer also gives you the option to buy a printed copy for $40, via "MyCopy Softcover" link on the right side of the webpage.


Homotopy theory:

Review of the fundamental group and Seifert--van Kampen theorem. Application to surfaces.
Homotopy and homotopy equivalence. CW complexes. Cellular approximation.
Category theory, functors and adjointness.
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 complexes, 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.


Lectures 1-4, Tu Jan 17 -- Th Jan 26: Review of homotopy theory and fundamental group. Retracts and homotopy retracts. Fundamental group and Seifert-van Kampen theorem. Examples of computations of the fundamental group. Surfaces and their fundamental groups. Mobius band and the Klein bottle. Surfaces and their fundamental groups. Lecture notes
Additional resources: R.Koch, Classification of surfaces       B.Guillou and H.Skiadas, Category theory

Lecture 5, Tu Jan 31: CW-complexes and Homotopy Extension Property (Hatcher, Chapter 0). Application to homotopy equivalences.


Homework will be assigned on Thursdays, due Thursday the following week. The numerical grade for the course will be the following linear combination: 35% homework, 5% quizzes, 20% midterm, 40% final. The lowest homework score will be dropped.

Additional online resources

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