Introduction to algebraic topology, fall 2011

Lecturer

Aleksandr Pasharin

Scope

10 cu.

Type

Advanced studies

Prerequisites

Topology II, Algebra I, Linear Algebra I.

Contents

The course is intended to be the first introduction to the singular homology theory and homological methods in algebraic topology.

We start off with the brief excursion to the world of simplices, simplicial methods and Delta-complexes. Then we move on to the main subject of the course - construction of the singular homology theory. We go through all the essential properties of the singular homology and apply it to prove the classical topological results such as the Invariance of Domain, Brouwer's fixed-point theorem, Brouwer-Jordan separation theorem, the main theorem of Algrebra and others.

Lecture Notes

Foreword
01 - Simplicial stuff
02 - Homological algebra
03 - Singular Homology
04 - CW complexes and cellular homology

Exercises

Exercises 1 Solutions 1
Exercises 2 Solutions 2
Exercises 3 Solutions 3
Exercises 4 Solutions 4
Exercises 5 Solutions 5
Exercises 6 Solutions 6
Exercises 7 Solutions 7
Exercises 8 Solutions 8
Exercises 9 Solutions 9
Exercises 10 Solutions 10
Exercises 11 Solutions 11
Exercises 12 Solutions 12

Lectures

Weeks 36-42 and 44-50, Wednesday 10-12 in room B321, Friday 11-13 in room B322. Two hours of exercise classes per week.

Exams

Final exam. Also bonus points for the exercises: 25% - 1 point, 40% - 2 points, 50% - 3 points, 60% - 4 points, 75% - 5 points.

Bibliography

Allen Hatcher: Algebraic Topology

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Exercise groups