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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:568570 |
| Date | January 2012 |
| Creators | Robinson, Daniel Mark |
| Contributors | Ray, Nige |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/the-homotopy-exponent-problem-for-certain-classes-of-polyhedral-products(a20ca2ac-540b-41e9-8af7-1f76d8f5ed84).html |
Page generated in 0.0134 seconds