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Wednesday, March 4, 2009 3:45 PM PHSC 809
Abstract Given a group G and two spaces X and Y with G actions, a fundamental question is whether there is a G-equivariant (i.e. f(gx) = gf(x) for all x in X) map (= continuous function) f from X to Y. For instance a classical theorem of Borsuk (1933) proving a conjecture of Ulam, states that there is no odd (i.e. f(-x) = -f(x) for all x) map from a sphere to a lower dimensional sphere. An odd map is a G-equivariant map where the group G here is cyclic of order two (acting antipodally on both spheres). In addition to its intrinsic beauty this result has remarkable utility with many consequences in diverse areas of mathematics including:
  • The Ham Sandwich Theorem (Banach-Steinhaus 1938, Stone-Tukey 1942 and a polynomial version by Gromov 2004).
  • Lovasz's proof of the Kneser Conjecture (1978),
  • The Topological Radon Theorem (Bajmoczy-Barany 1979).
  • Splitting Necklaces: Two Thieves (Alon-West 1986).
The relevant properties of the spaces and the group actions are more explicit in the following theorem of Dold (1983): Let G be cyclic of prime order acting freely (i.e. for each x in the space and all g in G gx = x only when g is the identity element) on X and Y. If X is dimY-connected then there is no equivariant map from X to Y. This result also is rich in applications such as:
  • The Topological Tverberg Theorem: k = prime (Barany-Shlossman-Szucs 1981).
  • Splitting Necklaces: N Thieves (Alon 1987).
However the freeness of the action is so strong a hypothesis that other potential applications (e.g. Topological Tverberg when k is not a prime) are ruled out. Of course some hypothesis on the action of G on Y is necessary to conclude non-existence of G-maps, if there is a point in Y fixed by every element of G then there will be an equivariant constant map from X to Y. Here is a generalization I am planning to sketch a proof of: Let G be a p-torus (i. e. a product of cyclic groups of order p, a fixed prime) acting on X and Y without G-fixed points. If X is mod-p acyclic up to the (homological) dimension of Y then there is no G-map from X to Y.

Unfortunately this result can not be generalized to an arbitrary (finite) p-group (even a cyclic one) as stated. I hope to give examples illustrating the difficulties when G is not a p-torus and partial results (especially for non-abelian p-groups) as time permits, but definitive such results for an arbitrary p-group are not available yet.