Homology with multiple-valued functions applied to fixed points

Abstract:

Certain multiple-valued functions (m-functions) are defined and a homology theory based upon them is developed. In this theory a singular simplex is an m-function from a standard simplex to a space and an m-function from one space to another induces a homomorphism of homology modules. In a family of functions ${f_x}:Y \to Y$ indexed by $x \in X$ the fixed points of ${f_x}$ are taken to be the images at x of a multiple-valued function $\phi :X \to Y$. In certain circumstances $\phi$ is an m-function, giving information about the behavior of the fixed points of ${f_x}$ as x varies over X. These facts are applied to self-maps of products of compact polyhedra and the question of whether such a product has the fixed point property for continuous functions is essentially reduced to the question of whether one of its factors has the fixed point property for m-functions. Some light is thrown on the latter problem by using the homology theory to prove a Lefschetz fixed point theorem for m-functions.