# 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, putyra@math.columbia.edu 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 E-mail: khovanov@math.columbia.edu Midterm: Thursday, March 14 Final exam: Take-home Webpage: www.math.columbia.edu/~khovanov/algtop2013

## Textbooks

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,
Index.

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

## Syllabus

Homotopy theory:

Homotopy and homotopy equivalence. CW complexes. Cellular approximation.
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

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
Midterm
Final exam