Fixed points of unitary decomposition complexes

Geometry Topology Seminar
Monday, November 18, 2013 - 14:00
1 hour (actually 50 minutes)
Skiles 006
For a fixed integer n, consider the nerve L_n of the topological poset of orthogonal decompositions of complex n-space into proper orthogonal subspaces. The space L_n has an action by the unitary group U(n), and we study the fixed points for subgroups of U(n). Given a prime p, we determine the relatively small class of p-toral subgroups of U(n) which have potentially non-empty fixed points. Note that p-toral groups are a Lie analogue of finite p-groups, thus if we are interested in the U(n)-space L_n at a fixed prime p, only the p-toral subgroups of U(n) play a significant role. The space L_n is strongly related to the K-theory analogues of the symmetric powers of spheres and the Weiss tower for the functor that assigns to a vector space V the classifying space BU(V). Our results are a step toward a K-theory analogue of the Whitehead conjecture as part of the program of Arone-Dwyer-Lesh. This is joint work with J.Bergner, R.Joachimi, K.Lesh, K.Wickelgren.