On decomposing suspensions of simplicial spaces

Abstract

Let $X_{\bullet}$ denote a simplicial space. The purpose of this note is to record a decomposition of the suspension of the individual spaces $X_n$ occurring in $X_{\bullet}$ in case the spaces $X_n$ satisfy certain mild topological hypotheses and where these decompositions are natural for morphisms of simplicial spaces. In addition, the summands of $X_n$ which occur after one suspension are stably equivalent to choices of filtration quotients of the geometric realization $|X_{\bullet}|$. The purpose of recording these decompositions is that they imply decompositions of the single suspension of certain spaces of representations as well as other varieties and are similar to decompositions of suspensions of moment-angle complexes which appear in a different context.