2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/124129It is shown that if T is a connected nontrivial graph and X is an arbitrary finite simplicial complex, then there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma. Along the way several results regarding Hom complexes, exponentials, and subdivision are established that may be of independent interest.14 pages, 12 figuresCombinatoricsAlgebraic Topology05C15, 55P15, 57M15text