WI-posets, graph complexes and Z2-equivalences

An evergreen theme in topological graph theory is the study of graph complexes, (Proof of the Lovász conjecture, arXiv:math.CO/0402395, 2, 2004; J. Combin. Theory Ser. A 25 (1978) 319–324; Using the Borsuk–Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry, Springer Universitext, Berlin, 2003; [17]). Many of these complexes are Z 2 -spaces and the associated Z 2 -index Ind Z 2 ( X ) is an invariant of great importance for estimating the chromatic numbers of graphs. We introduce WI-posets (Definition 2) as intermediate objects and emphasize the importance of Bredonʹs theorem (Theorem 9) which allows us to use standard tools of topological combinatorics for comparison of Z 2 -homotopy types of Z 2 -posets. Among the consequences of general results are known and new results about Z 2 -homotopy types of graph complexes. It turns out that, in spite of great variety of approaches and definitions, all Z 2 -graph complexes associated to G can be viewed as avatars of the same object, as long as their Z 2 -homotopy types are concerned. Among the applications are a proof that each finite, free Z 2 -complex is a graph complex and an evaluation of Z 2 -homotopy types of complexes Ind ( C n ) of independence sets in a cycle C n .