Decompositions of complex n-space into proper orthogonal subspaces form a (topological) poset with a natural action of the unitary group. I will discuss fixed point spaces of p-toral subgroups of U(n) on the nerve of this poset. I will show that most of them are actually contractible. This is recent progress in joint work with Bergner, Joachimi, Stojanoska, and Wickelgren.