This article gives a natural decomposition of the suspension of generalized moment-angle complexes or partial product spaces which arise as polyhedral product functors described below. The geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of Goresky-MacPherson concerning complements of certain subspace arrangements, as well as Davis-Januszkiewicz and Buchstaber-Panov concerning moment-angle complexes. Since the stable decompositions here are geometric, they provide corresponding homological decompositions for generalized moment-angle complexes for any homology theory.
We construct the Bousfield-Kan (unstable Adams) spectral sequence based on certain nonconnective periodic homology theories E such as complex periodic K -theory, and define an E -completion of a space X . For X = S 2 n +1 and E = K we calculate the E 2 -term and show that the spectral sequence converges to the homotopy groups of the K -completion of the sphere. This also determines all of the homotopy groups of the (unstable) K -theory localization of S 2 n +1 including three divisible groups in negative stems.
A combinatorial construction is used to analyze the properties of polyhedral products [1] and generalized moment-angle complexes with respect to certain operations on CW pairs including exponentiation. This allows for the construction of infinite families of toric manifolds, associated to a given one, in a way that simplifies the combinatorial input and, consequently, the presentation of the cohomology rings. The new input is the interaction of a purely combinatorial construction with natural associated geometric constructions related to polyhedral products and toric manifolds.
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