  For every simplicial complex K there exists a vertex-transitive simplicial
complex homotopy equivalent to a wedge of copies of K with some copies of the
circle. It follows that every simplicial complex can occur as a homotopy wedge
summand in some vertex-transitive complex. One can even demand that the
vertex-transitive complex is the clique complex of a Cayley graph or that it is
facet-transitive.
