Homotopy types of box complexes

In [14] Matousek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovasz's original bound [9] can be restated as chi(G) >= ind(B(G)) + 2. Sarkaria's bound [15] can be formulated as chi(G) >= ind(B-0(G)) + 1. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. The most interesting result is that UP to Z(2)-homotopy the box complex B(G) can be any Z(2)-space. This together with topological constructions allows us to construct graphs showing that the mentioned two bounds are different. Some of the results were announced in [14].