Indice de Point Fixe pour les Morphismes de Chaînes

2000 Mathematics Subject Classification: 54H25, 55M20.

The aim of this paper is to define a fixed point index for compact maps in the class of algebraic ANRs. This class, which we introduced in [2], contains all open subsets of convex subsets of metrizable topological vector spaces. In this class, it is convenient to study the fixed points of compact maps with the help of the chain morphisms that they induce on the singular chains. For this reason, we first define a fixed point index for a certain class of chain morphisms, and then define the fixed point index of compact maps as the fixed point index of the induced chain morphism. This fixed point index has all the usual properties of an index, including the mod p-theorem. The results of this paper are thus, in the metrizable case, a vast generalization of the Schauder conjecture.