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.
Date of Award | 1 Aug 2013 |
---|
Original language | English |
---|
Awarding Institution | - The University of Manchester
|
---|
Supervisor | Nigel Ray (Supervisor) |
---|
- Barratt's Conjecture
- Moore's Conjecture
- Moore space
- exponent
- homotopy
- polyhedral product
- moment angle complex
The Homotopy Exponent Problem for Certain Classes of Polyhedral Products
Robinson, D. (Author). 1 Aug 2013
Student thesis: Phd