Small models of graph colouring manifolds and the Stiefel manifolds Hom(C_5, K_n)

Abstract

We show P\'eter Csorba's conjecture that the graph homomorphism complex Hom(C_5,K_{n+2}) is homeomorphic to a Stiefel manifold, the space of unit tangent vectors to the n-dimensional sphere. For this a general tool is developed that allows to replace the complexes Hom(G, K_n) by smaller complexes that are homeomorphic to them whenever G is a graph for which those complexes are manifolds. The equivariant version of Csorba's conjecture is proved up to homotopy. We also study certain subdivisions of simplicial manifolds that are related to the interval poset of their face posets and their connection with geometric approximations to diagonal maps.