A fixed point conjecture for Borsuk continuous set-valued mappings

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.