Actions of symmetry groups

✍ Scribed by Graciela Chichilnisky

• Publisher: Springer
• Year: 1996
• Tongue: English
• Weight: 487 KB
• Volume: 13
• Category: Article
• ISSN: 0176-1714
• DOI: 10.1007/bf00179238

✦ Synopsis

This paper studies maps which are invariant under the action of the symmetry group Sk. 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 i proves that a full range map ~P : X k ~ X exists which is invariant under the action of Sk only if, for all i > 1, the elements of the homotopy group IIi(X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group Sk. Theorem 3 proves necessary and sufficient condition for a parafinite C W complex X to admit full range invariant maps for any prime number k : X must be contractible.