The University of Southampton
Courses

# MATH6137 Homotopy and Homology

## Module Overview

Students will gain an understanding of essential concepts in algebraic topology. They will be able to use logical and coherent arguments to prove basic results in homotopy theory, homology and cohomology. They will be able to apply and test the theoretical results on a range of spaces, including spheres, real and complex projective spaces, and matrix groups. Students will be able to demonstrate a practical grasp of the material by being able to calculate an array of algebraic invariants involving homotopy groups; simplicial, singular and cellular homology groups; and cup products in cohomology

### Aims and Objectives

#### Module Aims

Description Students will gain an understanding of essential concepts in algebraic topology. They will be able to use logical and coherent arguments to prove basic results in homotopy theory, homology and cohomology. They will be able to apply and test the theoretical results on a range of spaces, including spheres, real and complex projective spaces, and matrix groups. Students will be able to demonstrate a practical grasp of the material by being able to calculate an array of algebraic invariants involving homotopy groups; simplicial, singular and cellular homology groups; and cup products in cohomology.

### Syllabus

1. The Fundamental group and higher homotopy groups 2. Covering spaces and fibre bundles 3. Axiomatic homology theory 4. The equivalence of simplicial, singular and cellular homology 5. Examples and applications of homology 6. Cohomology 7. Cup products; applications of the cohomology ring 8. Time permitting: Poincare duality

### Learning and Teaching

#### Teaching and learning methods

Lectures and weekly problem sheets. Private study is also important for work on the problem sheets.

TypeHours
Independent Study150
Total study time150

Allen Hatcher (2001). Algebraic Topology.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Exam 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

CodeModule
MATH3086Galois Theory

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.