PMH1 Algebraic Topology
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.
Office: Carslaw 632
Timetable for Lectures: Monday 9am - 10pm and Wednesday 10am - 11pm in Room 830.
Consultation Hours: Monday 2 - 4pm, or by appointment.
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
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.
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.
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 . is due Wednesday, May 23. Examples of solutions: Example 1 , and for everything except question 2(c) Example 2 . Please submit copies directly either via email or in person; identical electronic copies also need to be submitted through Turnitin. In the event of special considerations, the maximum possible extension will be 7 days.
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.
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