Homotopy groups of Hom complexes of graphs

TL;DR

This paper explores the homotopy groups of Hom complexes of graphs, establishing a long exact sequence relating these groups to path graphs and providing a homotopy-theoretic framework for graph mappings.

Contribution

It introduces a long exact sequence connecting higher homotopy groups of pointed Hom complexes with path graph constructions, advancing the homotopy theory of graphs.

Findings

Established a long exact sequence relating homotopy groups of Hom complexes.

Proved an isomorphism between homotopy groups and homotopy classes of pointed graph maps.

Connected the homotopy theory of graphs with classical topological concepts.

Abstract

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.