Homotopy groups of Hom complexes of graphs

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.

💡 Research Summary

The paper develops a homotopy theory for graph maps within the category of pointed graphs by extending the notion of ×‑homotopy introduced in DocHom. The authors define Hom*(G,H) as the simplicial complex whose vertices are based graph homomorphisms from G to H, and they introduce H^I, the graph of based paths in H. By constructing natural maps between Hom*(G,H) and Hom*(G,H^I), they obtain a long exact sequence of homotopy groups that links the higher homotopy groups of Hom*(G,H) to those of Hom*(G,H^I). The central corollary shows that for every i ≥ 1, the i‑th homotopy group π_i(Hom*(G,H)) is naturally isomorphic to the set