Topological obstructions to graph colorings

For any two graphs $G$ and $H$ Lovász has defined a cell complex $\text{\tt Hom}\, (G,H)$ having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with $G$ a cycle of odd length. More specifically, we show that

If $\text{\tt Hom}\,(C_{2r+1},G)$ is $k$-connected, then $\chi(G)\geq k+4$.

Our actual statement is somewhat sharper, as we find obstructions already in the nonvanishing of powers of certain Stiefel-Whitney classes.