Department of Mathematics & Computer Science

MA4101 Algebraic Topology

Explanation of Pre-requisites

This module will draw on some basic ideas from both algebra and analysis. In algebra familiarity with the basic concepts of vector spaces, as covered in MA2102, will also be assumed, as will some of the elementary ideas of abstract algebra such as groups, rings or modules, sub- and quotient objects, and so on, as discussed in MA2111 or MA2161. In analysis the central topics drawn on are those of topological spaces, of closed or compact subsets of and continuous functions. These are all covered in MA2151, but their presentation and the examples in MA3011 will probably be helpful in a number of ways. This module would make a good follow-on module from MA3011.

Course Description

Fact: if you have a dog which is completely covered in hair, then there is no way of combing that hair smooth so that there is no parting or bald spot. This is the so-called `hairy dog theorem'.

Fact: no matter how badly you make a sandwich out of two pieces of bread and a slice of ham then it is always possible to find a plane cutting the sandwich which bisects exactly each piece of bread and the slice of meat. This is the so-called `ham sandwich theorem'.

Fact: if you associate to each point of the Earth's surface the two numbers where is the temperature at that point and is the air pressure there, then there is always at least one point which has the same values of and as its diametrically opposite point. This is the `Borsuk-Ulam theorem'.

These are all very deep geometric facts about the shapes of dogs, ham sandwiches and the Earth; they are especially deep as they are true for any shape or size of dog, sandwich, or planet, and they are all proved by a remarkably successful set of mathematical ideas and methods that date from early last century. These ideas, and ones like them, constitute the subject of Algebraic Topology.

The basic idea of the subject is to find a formal way of translating geometric problems into an appropriate algebraic language. If this is done successfully then the geometric problem is usually reduced to a fairly simple piece of algebra and the problem is solved or the theorem proved by relatively trivial algebra. In this module we shall prove each of the facts above by translating the underlying geometry into questions about integers, groups or vector spaces.

Aims

This module aims to introduce the basic ideas of algebraic topology and to demonstrate its power by proving some memorable theorems.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To be able to understand, reproduce and apply the main results and proofs in this module.

To be able to solve some routine problems in low and high dimensional topology.

Transferable Skills

The ability to present arguments and solutions in a coherent and logical form.

The ability to use the techniques taught within the module to solve problems.

The ability to apply taught principles and concepts to new situations.

Syllabus

Homology theory and homological algebra: elementary categorical concepts - category, functor. Topological spaces, homotopy equivalent spaces, polyhedra and the simplicial approximation theorem. Examples of standard spaces. Simplical and singular homology groups, their functorial properties and computation (the Eilenberg-Steenrod axioms). Relation to the fundamental group. Application of homology theory to Brouwer's fixed point theorem, the Ham Sandwich theorem, the hairy dog theorem and the Borsuk Ulam theorem. Other theorems as time allows.

Reading list

Recommended:

Background:

Details of Assessment

There will be six questions on the examination paper; any number of questions may be attempted but full marks may be gained from answers to four questions. All questions will carry equal weight.

Author: G. T. Laycock, tel: +44 (0)116 252 3902
Last updated: 2002-10-25
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