The Homotopy Exponent Problem for Certain Classes of Polyhedral Products PDF Download

Author: Daniel Mark Robinson
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Languages : en
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Book Description
Given a sequence of n topological pairs (X-i,A-i) for i=1,...,n, and a simplicial complex K, on n vertices, there is a topological space (X,A) K by a construction of Buchstaber and Panov. Such spaces are called polyhedral products and they generalize the central notion of the moment-angle complex in toric topology. We study certain classes of polyhedral products from a homotopy theoretic point of view. The boundary of the 2-dimensional n-sided polygon, where n is greater than or equal to 3, may be viewed as a 1-dimensional simplicial complex with n vertices and n faces which we call the n-gon. When K is an n-gon for n at least 5, (D 2,S 1) K is a hyperbolic space, by a theorem of Debongnie. We show that there is an infinite basis of the rational homotopy of the based loop space of (D 2,S 1) K represented by iterated Samelson products. When K is an n-gon, for n at least 3, and P m(p r) is a mod p r Moore space with m at least 3 and r at least 1, we show that the order of the elements in the p-primary torsion component in the homotopy groups of (Cone X, X) K, where X is the loop space of P m(p r), is bounded above by p {r+1}. This result provides new evidence in support of a conjecture of Moore. Moreover, this bound is the best possible and in fact, if a certain conjecture of M.J Barratt is assumed to be true, then this bound is also valid, and is the best possible, when K is an arbitrary simplicial complex.