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MA4101 Algebraic Topology

MA4101 Algebraic Topology

Credits: 20 Convenor: Prof. J.R. Hunton Semester: 2 (weeks 15 to 26)
Prerequisites: essential: MA2102 or MA2111, MA2151 desirable: MA3151
Assessment: Coursework: 10% Examination: 90%
Lectures: 36 Problem Classes: 6
Tutorials: none Private Study: 108
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Subject Knowledge

Aims

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

Learning Outcomes

Students will understand some of the common processes of translating topological or geometric infomation into algebraic information, seeing examples of this through singular and simplicial homology. They will gain some understanding of the language and techniques of category theory and homological algebra. They will know some of the classical applications of the algebraic topology such as the Ham Sandwich theorem, the Hairy Dog theorem the Borsuk-Ulam theorem.

Methods

Lectures, example classes, example sheets.

Assessment

Marked problem sheets, written examination.

Subject Skills

Aims

To see how one can use a number of apparently distinct branches of mathematics to solve new types of problem. To further increase problem solving and written presentation skills.

Learning Outcomes

Students will become familiar with one of the famous methods of translating of geometric/topological problems into algebraic ones as a means of solution. They will have knowledge of the ideas oand methods of Algebraic Topology, and will continue practice in problem solving, conceptual thinking, logical argument and written presentation.

Methods

Lectures, examples classes, example sheets.

Assessment

Marked problem sheets, written examination.

Explanation of Pre-requisites

This module needs some knowledge of both analysis and algebra. The analysis needed requires the student to be comfortable with discussing reasonably straightforward topological spaces such as spheres, tori, discs and so on, together with the idea of continuous functions between them. This is all covered in the Abstract Analysis course MA2151, but more familiarity with these conceps will be gained from the Topology module MA3151 which would be a good partner with this module.

The algebra used is a combination of elementary group theory and/or linear algebra (the student is free to choose which type of algebra they prefer, though ideally they should be familiar with the elementary concepts of both).

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 $(t,p)$ where $t$ is the temperature at that point and $p$ is the air pressure there, then there is always at least one point which has the same values of $t$ and $p$ 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 to mid 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 can be solved by algebraic means. In this module we shall prove each of the facts above by translating the underlying geometry into questions about integers, groups or vector spaces.

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:

M. A. Armstrong, Basic Topology, Springer.

Background:

W. S. Massey, A Basic Course in Algebraic Topology, Springer. H. Sato, Algebraic Topology: an intuitive approach, Springer.

Resources

Problem sheets, lecture rooms, lecturer.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA4111 Differential Geometry Up: ModuleGuide03-04 Previous: MA4041 Methods in Molecular Simulation

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Last updated: 2004-02-21
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