Homotopy groups of Hom complexes of graphs |
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Authors: | Anton Dochtermann |
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Institution: | Institut für Mathematik, MA 6-2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany |
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Abstract: | The notion of ×-homotopy from Anton Dochtermann, Hom 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 Hom∗(G,H) with the homotopy groups of Hom∗(G,HI). 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 πi(Hom∗(G,H))≅×G,ΩiH], where ΩH is the graph of based closed paths in H and ×G,K] is the set of ×-homotopy classes of pointed graph maps from G to K. This is similar in spirit to the results of Eric Babson, Hélène 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. |
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Keywords: | Graph homomorphisms Pointed graphs Hom complex Discrete homotopy Paths |
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