The complex of injective words of permutations which are not derangements is contractible

Abstract

Let $D_n\subseteq {\Sigma }_n$ be the set of derangements in the symmetric group. We prove that the complex of injective words generated by ${\Sigma }_n\setminus D_n$ is contractible. This gives a conceptual explanation to the well known fact that the complex of injective words generated by ${\Sigma }_n$ is homotopy equivalent to the wedge sum $\underset{|D_n|}{\bigvee }{S}^{n-1}$.