Simplicial Approximation

Abstract

We turn now to the problem of comparing polyhedra by means of their associated homology groups. Comparisons between two topological spaces are usually made on the basis of a continuous map, ideally a homeomorphism, from one space to another. Groups are compared by means of homomor-phisms and isomorphisms. We shall show in this chapter that a continuous map f: |K| → |L| induces for each non-negative integer p a homomorphism f* _p : H _P (K) → H _P (L) on the associated homology groups. This will allow topological comparisons between the polyhedra |K| and |L| on the basis of algebraic similarities between their associated homology groups.