The universality of Hom complexes

2007/02/16 by Anton Dochtermann, Dochtermann, Anton
Chemistry · Computer Science · Mathematics · #05C15 #55P15 #57M15 #Algebraic Topology (math.AT) #Axial and Atropisomeric Chirality Synthesis #Combinatorics (math.CO) #FOS: Mathematics #Molecular spectroscopy and chirality #Topological and Geometric Data Analysis #math.AT #math.CO #msc:05C15 #msc:55P15 #msc:57M15

paper · pdf · doi:10.48550/arxiv.math/0702471

14 pages, 12 figures

arxiv created 2007/02/16 · openalex publication_date 2007/02/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It 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.

Related

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• Homotopy types of Hom complexes of graph homomorphisms whose codomains are cycles
• Hom complexes of graphs whose codomains are square-free
• Hom complexes and homotopy theory in the category of graphs
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