Small models of graph colouring manifolds and the Stiefel manifolds

We show Péter Csorbaʹs 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 Csorbaʹs conjecture is proved up to homotopy. o 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.