Actions of symmetry groups

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 :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 :X ^k X exists which is invariant under the action of S _k only if, for all i1, the elements of the homotopy group _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.Hospitality and research support from the Standard Institute for Theoretical Economics during the summer of 1991 is gratefully acknowledged. This paper was presented at a Colloquium in the Department of Mathematics, Columbia University, February 6, 1991. I thank the participants of the Colloquium and Jerry Kelly for helpful comments.