Homotopy groups of Hom complexes of graphs

摘要

The notion of x-homotopy from [Anton Dochtermann, Horn complexes and homotopy theory in the category of graphs, European J. Combin., in press] is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space Horn,(G. H) with the homotopy groups of Hom(*) (G, H-1). Here Hom(*)(G, H) is a space which parameterizes pointed graph maps from G to H (a pointed version of the usual Hom complex), and HI is the graph of based paths in H. As a corollary it is shown that pi(i)(Hom(*)(G, H)) congruent to [G, Omega H-i](x), where Omega H is the graph of based closed paths in H and [G, K](x) is the set of x-homotopy classes of pointed graph maps from G to K. This is similar in spirit to the results of [Eric Babson, son, Helene Barcelo, Mark de Longueville, Reinhard Laubenbacher, Homotopy theory of graphs, J. Algebraic Combin. 24 (1) (2006) 31-44], where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed. (C) 2008 Elsevier Inc. All rights reserved.