Introductory description

Algebraic topology is concerned with the construction of algebraic invariants (usually groups) associated to topological spaces which serve to distinguish between them. Most of these invariants are ``homotopy'' invariants. In essence, this means that they do not change under continuous deformation of the space and homotopy is a precise way of formulating the idea of continuous deformation. This module will concentrate on constructing the most basic family of such invariants, homology groups, and the applications of these homology groups.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The starting point will be simplicial complexes and simplicial homology. An n-simplex is the n-dimensional generalisation of a triangle in the plane. A simplicial complex is a topological space which can be decomposed as a union of simplices. The simplicial homology depends on the way these simplices fit together to form the given space. Roughly speaking, it measures the number of p-dimensional "holes'' in the simplicial complex. For example, a hollow 2-sphere has one 2-dimensional hole, and no 1-dimensional holes. A hollow torus has one 2-dimensional hole and two 1-dimensional holes. Singular homology is the generalisation of simplicial homology to arbitrary topological spaces. The key idea is to replace a simplex in a simplicial complex by a continuous map from a standard simplex into the topological space. It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition. One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes. This result means that we can combine the theoretical power of singular homology and the computability of simplicial homology to get many applications. These applications will include the Brouwer fixed point theorem, the Lefschetz fixed point theorem and applications to the study of vector fields on spheres.

Learning outcomes

By the end of the module, students should be able to:

• Give the definitions of simplicial complexes and their homology groups and a geometric understanding of what these groups measure
• Give techniques for computing these groups
• Give the extension to singular homology
• Understand the theoretical power of singular homology
• Develop a geometric understanding of how to use these groups in practice

Indicative reading list

There is no book which covers the module as it will be taught. However, there are several books on algebraic topology which cover some of the ideas in the module, for example: JW Vick, Homology Theory, Academic Press. MA Armstrong, Basic Topology, McGraw-Hill. Additional references: CRF Maunder, Algebraic Topolgy, CUP. A Dold, Lectures on Algebraic Topology, Springer-Verlag. C Kosniowski, A first course in algebraic topology, CUP. MJ Greenberg and JR Harper, Algebraic Topology: A first course, Addison-Wesley.

Subject specific skills

By the end of the module the student should be able to develop a geometric understanding of how to use these groups in practice

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.