A geometric approach to the fixed point index

Abstract

J. Leray defined a local fixed point index for functions defined in what he called convexoid spaces. From the standpoint of analysis, the most important example of a convexoid space is a compact subset C c X, X a locally convex topological vector space, such that C = U ?=1 d, where d are compact, convex subsets of X or a homeomorphic image of such a C. In this paper a simple geometric approach is given (see Lemma 2 below) by means of which a fixed point index can be defined for functions with domain in a class of spaces ^ which contains the spaces C mentioned above and also the compact metric ANR's. The usual properties of the fixed point index are established, and it is shown that they axiomatically determine the index for the class of spaces J^~.