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 → Y {f_x}:Y \to Y indexed by x ∈ X x \in X the fixed points of f x {f_x} are taken to be the images at x of a multiple-valued function ϕ : X → Y \phi :X \to Y . In certain circumstances ϕ \phi is an m-function, giving information about the behavior of the fixed points of f x {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.