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Undergraduate Study

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 Tuesday 9am - 10pm in Room 830.

Consultation Hours: Monday 1 - 4pm, or by appointment.

Exam: Monday 19 June, 2 - 4:10 pm, AGR.

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
  • Weeks 3-7: Chapter 1: Fundamental group
  • Weeks 8-12: Chapter 2: Homology
  • Week 13: wrap up

Lecture Notes

The lecture notes will be completed and updated during the semester. They 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.
2. A written exam worth 60% to be held at the end of semester 1, covering the whole content of the course. The exam time and venue will be anounced later.

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.

Timetable

 

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