A simple proof for folds on both sides in complexes of graph homomorphisms

In this paper we study implications of folds in both parameters of Lovász' Hom$(-,-)$ complexes. There is an important connection between the topological properties of these complexes and lower bounds for chromatic numbers. We give a very short and conceptual proof of the fact that if $G-v$ is a fold of $G$, then ${bd}$Hom$(G,H)$ collapses onto ${bd}$Hom$(G-v,H)$, whereas Hom$(H,G)$ collapses onto Hom$(H,G-v)$.

We also give an easy inductive proof of the only nonelementary fact which we use for our arguments: if $\varphi$ is a closure operator on $P$, then $\Delta(P)$ collapses onto $\Delta(\varphi(P))$.