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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 713

Email: "first name" . "last name" @sydney.edu.au

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

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

Exam: Tuesday June 18 (9 - 11:10 am) in AGR (Carslaw 829). You are not allowed to use any book, but you may use one prepared double-sided sheet of paper.

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 theory of algebraic topology constructs and studies 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, will be posted here two weeks before the due date.

Please submit the assignments through Turnitin. In the event of special considerations, the maximum possible extension will be 7 days.

2. The written exam will be worth 60% and will cover the whole content of the course.

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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