ACTIONS OF SYMMETRY GROUPS

This paper studies maps which are invariant under the action of the sy mmetry group S-k. The problem originates in social choice theory: ther e are k individuals each with a space of preferences X, and a social c hoice map Phi:X(k) --> X which is anonymous i.e. invariant under the a ction of a group of symmetries. Theorem 1 proves that a full range map Psi:X(k) --> X exists which is invariant under the action of S-k only if, for all i greater than or equal to 1, the elements of the homotop y group Pi(i)(X) have orders relatively prime with k. Theorem 2 derive s a similar results for actions of subgroups of the group S-k. Theorem 3 proves necessary and sufficient condition for a parafinite CW compl ex X to admit full range invariant maps for any prime number k:X must be contractible.