Proof of the Lovász conjecture

To any two graphs G and H one can associate a cell complex Horn (G, H) by taking all graph multihomomorphisms from G to H as cells. In this paper we prove the Lovasz conjecture which states that if Horn (C 2r+1 , G) is k-connected, then Χ(G) > k + 4, where r, k ∈ Z, r > 1, k > -1, and C 2r+1 denotes the cycle with 2r+1 vertices. The proof requires analysis of the complexes Horn (C 2r+1 , K n ). For even n, the obstructions to graph colorings are provided by the presence of torsion in H* (Hom (C 2r+1 , K n ); Z). For odd n, the obstructions are expressed as vanishing of certain powers of Stiefel-Whitney characteristic classes of Horn (C 2r+1 , K n ), where the latter are viewed as Z 2 -spaces with the involution induced by the reflection of C 2r+1 .