Computer Graphics and Geometric Modelling (PDF)

Chapter 7
Algebraic Topology (p. 358)

7.1 Introduction

The central problem of algebraic topology is to classify spaces up to homeomorphism by means of computable algebraic invariants. In the last chapter we showed how two invariants, namely, the Euler characteristic and orientability, gave a complete classi- .cation of surfaces. Unfortunately, these invariants are quite inadequate to classify higher-dimensional spaces. However, they are simple examples of the much more general invariants that we shall discuss in this chapter.

The heart of this chapter is its introduction to homology theory. Section 7.2.1 de.nes the homology groups for simplicial complexes and polyhedra, and Section 7.2.2 shows how continuous maps induce homomorphisms of these groups. Section 7.2.3 describes a few immediate applications. In Section 7.2.4 we indicate how homology theory can be extended to cell complexes and how this can greatly simplify some computations dealing with homology groups. Along the way we de.ne CW complexes, which are really the spaces of choice in algebraic topology because one can get the most convenient description of a space with them. Section 7.2.5 de.nes the incidence matrices for simplicial complexes.

These are a fundamental tool for computing homology groups with a computer. Section 7.2.6 describes a useful extension of homology groups where one uses an arbitrary coef.cient group, in particular, Z2. After this overview of homology theory we move on to de.ne cohomology in Section 7.3. The cohomology groups are a kind of dual to the homology groups.

We then come to the other major classical topic in algebraic topology, namely, homotopy theory. We start in Sections 7.4.1 and 7.4.2 with a discussion of the fundamental group of a topological space and covering spaces. These topics have their roots in complex analysis. Section 7.4.3 sketches the de.nition of the higher-dimensional homotopy groups and concludes with