Document Zbl 1228.55007 - zbMATH Open
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On decomposing suspensions of simplicial spaces. (English) Zbl 1228.55007
Let \(X_\bullet\) be a simplicial space with individual spaces \(X_n\). The structure maps \(s_j : X_n \to X_{n+1}\) induce filtrations
\[
S^n(X_n) = s_0^n(X_0) \subset \cdots \subset S^t(X_n) \subset \cdots \subset S(X_n) \subset S^0(X_n) = X_n.
\]
Under the assumption that \((S^{t-1}(X_n), S^t(X_n))\) is an NDR-pair for all \(n\) and \(t \geq 1\), the authors prove that these filtrations are split up to homotopy after suspension. Further, the corresponding decompositions of \(\Sigma (X_n)\) are natural with respect to morphisms of simplicial spaces and the summands are stably equivalent to the quotients in the filtration \(F_j|X_\bullet|\) of the geometric realization of \(X_\bullet\) given in [J. P. May, The geometry of iterated loop spaces. Lecture Notes in Mathematics. 271. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.55009)]. The result applies to a variety of important examples. These include the simplicial space of commuting \(n\)-tuples in a Lie group, moment-angle complexes and the simplicial space arising from a compact real algebraic variety.