# PMH1 Algebraic Topology

## General Information

This page relates to the Pure Mathematics Honours course "Algebraic Topology".

Lecturer for this course: Kevin Coulembier.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.

## Organisational Matters

**Office:** Carslaw 632

**Email:** kevin.coulembier@sydney.edu.au

**Timetable for Lectures:** *Monday 9am - 10pm* and *Wednesday 10am - 11pm* in Room **830**.

**Consultation Hours:** Monday 2 - 4pm, or by appointment.

** Exam:**

## Overview

The theory of topology is concerned with the
properties of space that are preserved under continuous deformations (stretching, crumpling and
bending), but not tearing or gluing.

The basic goal of algebraic topology is to find algebraic invariants (groups, rings, algebras, modules) that allow to distinguish between certain topological spaces. We will see the basics of two such
algebraic invariants, namely homotopy groups and homology groups.

- (Weeks 1-2) Chapter 0:
**Introduction.**Motivation for algebraic topology, basic notions of topology and homotopy theory, cell complexes.

- (Weeks 3-7) Chapter 1:
**Fundamental group.**Definition of the fundamental group, invariance under homotopy equivalences, van Kampen's theorem, covering spaces, deck transformations.

- (Weeks 8-12) Chapter 2:
**Homology.**Basic notions of homological algebra, singular homology, relative homology, simplicial homology, cellular homology.

- (Week 13) wrap up

## Lecture Notes

The lecture notes are only meant to accompany the book by Hatcher. The exact course content will consist of everything mentioned in the lecture notes (except the parts marked SI, which consist of supplementary information) and all parts of the book which relate to the notes.

## Exercises

The lecture notes contain a lot of short exercises, meant to test understanding of definitions or to fill in small gaps in the exposition of the theory. The lecture notes of Chapters 1 and 2 also contain lists of recommended exercises in Hatcher's book, for each section. These are more advanced exercises.

## Assessment

1. Two assignments worth 20% each; they will be posted here two weeks before the due date.

- The first assignment is due Wednesday, April 18. Examples of solutions: Example 1, Example 2 and Example 3.

- The second assignment is due Wednesday, May 23. Examples of solutions: Example 1, and for everything except question 2(c) Example 2.

2. The written exam worth 60%, covering the whole content of the course, will take place on Wednesday June 20, 1:20-3:30pm in AGR (Carslaw 829). Eeach student is allowed to use a prepared two-sided page of notes during the exam.

## References

The main reference will be

A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. ISBN: 0-521-79160-X

Some errors in the original version have been corrected in the online version.