On decomposing suspensions of simplicial spaces

TL;DR

This paper presents a natural decomposition of the suspension of each space in a simplicial space under mild conditions, linking these decompositions to stable equivalences of filtration quotients of the geometric realization.

Contribution

It introduces a natural decomposition method for suspensions of simplicial space components, connecting them to filtration quotients of the geometric realization.

Findings

Decomposition of suspensions of $X_n$ under mild hypotheses.

Stable equivalence of summands to filtration quotients of $|X_{ullet}|$.

Applications to decompositions of representation spaces and moment-angle complexes.

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.