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Polyhedral products and features of their homotopy theory. (English) Zbl 1476.55024

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 103-144 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: This chapter reviews the various fundamental unstable and stable splitting theorems for the polyhedral product. It presents results on the cohomology of polyhedral products. The chapter describes the application of polyhedral products to questions concerning the Golod properties of certain rings. It discusses higher Whitehead products, constructed using polyhedral products. A polyhedral product is a natural topological subspace of a Cartesian product, determined by a simplicial complex \(K\) and a family of pointed pairs of spaces, one for each vertex of \(K\). The development of the theory of polyhedral products was guided by their inextricable link to spaces known as moment-angle manifolds which arose within the subject of toric topology. Higher Whitehead products were introduced into the homotopy theory of moment-angle complexes in the work of T. Panov and N. Ray.
For the entire collection see [Zbl 1468.55001].

MSC:

55P42 Stable homotopy theory, spectra
55Q15 Whitehead products and generalizations
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
52B11 \(n\)-dimensional polytopes
32S22 Relations with arrangements of hyperplanes
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14F45 Topological properties in algebraic geometry
55U10 Simplicial sets and complexes in algebraic topology
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55N10 Singular homology and cohomology theory
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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