The Universality of Hom Complexes

@inproceedings{Dochtermann2007TheUO,
  title={The Universality of Hom Complexes},
  author={Anton Dochtermann},
  year={2007}
}
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. 

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