Homotopy groups of Hom complexes of graphs

Details Homotopy groups of Hom complexes of graphs Homotopy groups of Hom complexes of graphs

2007-05-18

arxiv.org/abs/0705.2620v3

Mathematics

Combinatorics

20 pages, 6 figures, final version, to be published in J. Combin. Theory Ser. A

Scientific paper

The notion of $\times$-homotopy from \cite{DocHom} 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 $\Hom_*(G,H^I)$. Here $\Hom_*(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $\Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that $\pi_i \big(\Hom_*(G,H) \big) \cong [G,\Omega^i H]_{\times}$, where $\Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{\times}$ is the set of $\times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of \cite{BBLL}, 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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