Transitivity is not a (big) restriction on homotopy types

TL;DR

The paper demonstrates that any simplicial complex can be embedded as a homotopy wedge summand in a vertex-transitive complex, showing transitivity restrictions are not significant for homotopy types.

Contribution

It constructs vertex-transitive simplicial complexes homotopy equivalent to wedges of given complexes, including special cases like Cayley graph clique complexes.

Findings

Any simplicial complex can be realized as a homotopy wedge summand in a vertex-transitive complex.

Vertex-transitive complexes can be chosen to be clique complexes of Cayley graphs.

Transitivity restrictions do not significantly limit the homotopy types realizable in such complexes.

Abstract

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.