A Furi-Pera theorem in Hausdorff topological spaces for acyclic maps

We present new Furi-Pera theorems for acyclic maps between topological spaces. 1. Introduction. In this paper, we present new Furi-Pera theorems [6, 7] for acyclic maps between Hausdorff topological spaces. The main result in our paper is based on a new Leray-Schauder alternative [1] for such maps which in turn is based on the notion of compactly null-homotopic. We first recall some results and ideas from the literature. Let X and Z be subsets of Hausdorff topological spaces. We will consider maps F : X → K(Z); here K(Z) denotes the family of nonempty compact subsets of Z. A nonempty topological space is said to be acyclic if all its reduced ˘ Cech homology groups over the rationals are trivial. Now F : X → K(Z) is acyclic if F is upper semicontinuous with acyclic values. Suppose X and Z are topological spaces. Given a class ᐄ of maps, ᐄ(X, Z) denotes the set of maps F : X → 2 Z (nonempty subsets of Z) belonging to ᐄ, and ᐄ c the set of finite compositions