A fixed point conjecture for Borsuk continuous set-valued mappings

Abstract

Abstract. The main result of this paper is that for n = 3, 4, 5 and k = n − 2, every Borsuk continuous set-valued map of the closed ball in the n-dimensional Euclidean space with values which are one-point sets or sets homeomorphic to the k-sphere has a fixed point. Our approach fails for (k, n) = (1, 4). A relevant counterexample (for the homological method, not for the fixed point conjecture) is indicated.