Content: 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.
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.
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 it is quite hard to compute singular homology 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 computational power 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.
Aims: To introduce homology groups for simplicial complexes; to extend these to the singular homology groups of topological spaces; to prove the topological and homotopy invariance of homology; to give applications to some classical topological problems.
Objectives: To give the definitions of simplicial complexes and their homology groups and a geometric understanding of what these groups measure; to give techniques for computing these groups; to give the extension to singular homology; to understand the theoretical power of singular homology; to develop a geometric understanding of how to use these groups in practice.
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.
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.