 
Summary: ON DECOMPOSING SUSPENSIONS OF SIMPLICIAL SPACES
A. ADEM, A. BAHRI, M. BENDERSKY, F. R. COHEN, AND S. GITLER
Abstract. Let X· denote a simplicial space. The purpose of this note is to record a de
composition of the suspension of the individual spaces Xn occurring in X· in case the spaces
Xn satisfy certain mild topological hypotheses and where these decompositions are natural
for morphisms of simplicial spaces. In addition, the summands of Xn which occur after one
suspension are stably equivalent to choices of filtration quotients of the geometric realization
X·. The purpose of recording these decompositions is that they imply decompositions of
the single suspension of certain spaces of representations [1, 2] as well as other varieties and
are similar to decompositions of suspensions of momentangle complexes [4] which appear
in a different context.
1. Introduction and Statement of Results
Let X· denote a simplicial space. The purpose of this note is to give a decomposition of
the suspension of the individual spaces Xn occurring in X· in case the spaces Xn satisfy
certain mild topological hypotheses. These decompositions are natural for morphisms of
simplicial spaces. In addition, the summands of Xn which occur after one suspension are
stably equivalent to choices of filtration quotients of the geometric realization X·.
These structures occur in several contexts in useful ways and the following spaces admit
decompositions of the type discussed above.
(1) The suspension of the the loop space for a (pathconnected) suspension of a CW
