Hom complexes of graphs of diameter $1$

Anurag Singh, Nandini Nilakantan

Published 2018-07-27Version 1

Given a finite, simplicial complex $X$ and a connected graph $T$ with diameter $1$, in this article, we show that $\text{Hom}(T, G_{1, X})$ is homotopy equivalent to $X.$ Here, $G_{1,X}$ is the reflexive graph obtained by taking the $1$-skeleton of the first barycentric subdivision of $X$ and adding a loop at each vertex. This problem was proposed by Dochtermann in \cite{anton}.