Actions of symmetry groups

Abstract

This paper studies maps which are invariant under the action of the symmetry group S k . The problem originates in social choice theory: there are k individuals each with a space of preferences X, and a social choice map ϕ:X k → X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map ψ:X k → X exists which is invariant under the action of S k only if, for all i≥1, the elements of the homotopy group Π i (X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group S k . Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k:X must be contractible.