Homology and Fixed Points

Abstract

In this chapter we develop the algebraic and geometric notions needed to formulate and prove the main result, the Lefschetz-Hopf theorem for polyhedra. We further illustrate the use of homology by studying the special case of maps S ⁿ → S ⁿ, showing that the Brouwer degree of a map not only completely characterizes its homotopy behavior, but also gives considerable information about special topological features that such a map may have. We come full circle with the beginning of the last chapter by deriving Borsuk’s antipodal theorem within this homological framework.