The University of Southampton
Courses

# MATH3080 Algebraic Topology

## Module Overview

Topology is concerned with the way in which geometric objects can be continuously deformed to one another. It can be thought of as a variation of geometry where there is a notion of points being "close together" but without there being a precise measure of their distance apart. Examples of topological objects are surfaces which we might imagine to be made of some infinitely malleable material. However much we try, we can never deform in a continuous way a torus (the surface of a bagel) into the surface of the sphere. Other kinds of topological objects are knots, i.e. closed loops in 3-dimensional space. Thus, a trefoil or "half hitch" knot can never be deformed into an unknotted piece of string. It's the business of topology to describe more precisely such phenomena. In topology, especially in algebraic topology, we tend to translate a geometrical, or better said a topological problem to an algebraic problem (more precisely, for example, to a group theoretical problem). Then we solve that algebraic problem and try to see what that solution tells us of our initial topological problem. So to do topology you need to work equally well with both geometric and algebraic objects.

### Aims and Objectives

#### Module Aims

The primary aim of this module is to explore properties of topological spaces. We shall consider in detail examples such as surfaces. To distinguish topological spaces, we need to define topological invariants, such as the "fundamental group" or the "homology" of a space". To enable us to do this, knowledge of basic group theory and topology is essential. Some background in real analysis would also be helpful.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Understand and apply the notion of homotopy and compute homotopy groups for some examples
• Understand and apply the notion of homology and compute homology groups for some examples..

### Syllabus

Fundamental group: • Homotopy and homotopy type • Paths and homotopy, definition fundamental group, simply connected spaces. • Fundamental group in general: induced homomorphisms, calculations and applications (R^2 is not homeomorphic to R^n for n not equal to 2.) • Covering maps, covering spaces, universal cover, deck transformations. Homology: • Simplical complexes. • Simplicial and singular homology. • Homotopy invariance. • Exact sequences and excision, applications: Brouwer fixed point theorem, distinguishing spheres. • Seifert-van Kampen Theorem. If time permits one or more of the following topics: • Euler characteristic • Lefschetz fixed point theorem

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem sheets, private study

TypeHours
Independent Study150
Total study time150

A. HATCHER. Algebraic topology.

M.A. ARMSTRONG. Basic Topology.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Exam 80%

#### Referral

MethodPercentage contribution
Exam %

#### Pre-requisites

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

CodeModule
MATH2003Group Theory
MATH2039Analysis
MATH3079Metric Spaces and Topology

### 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.