The homotopy type of the polyhedral product for shifted complexes
The homotopy type of the polyhedral product for shifted complexes
We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices,X1,...,Xn are pointed connected CW-complexes and CXi is the cone on Xi, then the polyhedral product determined by K and the pairs (C Xi , Xi ) is homotopy equivalent to a wedge of suspensions of smashes of the Xi ’s. Earlier work of the authors dealt with the special case where each Xi is a loop space. New techniques are introduced to prove the general case. These have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes. We discuss connections between polyhedral products and toric topology, combinatorics, and classical homotopy theory.
davis–januszkiewicz space, moment–angle complex, polyhedral product, shifted complex, homotopy type
690-715
Grbić, Jelena
daaea124-d4cc-4818-803a-2b0cb4362175
Theriault, Stephen
5e442ce4-8941-41b3-95f1-5e7562fdef80
1 October 2013
Grbić, Jelena
daaea124-d4cc-4818-803a-2b0cb4362175
Theriault, Stephen
5e442ce4-8941-41b3-95f1-5e7562fdef80
Grbić, Jelena and Theriault, Stephen
(2013)
The homotopy type of the polyhedral product for shifted complexes.
Advances in Mathematics, 245, .
(doi:10.1016/j.aim.2013.05.002).
Abstract
We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices,X1,...,Xn are pointed connected CW-complexes and CXi is the cone on Xi, then the polyhedral product determined by K and the pairs (C Xi , Xi ) is homotopy equivalent to a wedge of suspensions of smashes of the Xi ’s. Earlier work of the authors dealt with the special case where each Xi is a loop space. New techniques are introduced to prove the general case. These have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes. We discuss connections between polyhedral products and toric topology, combinatorics, and classical homotopy theory.
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e-pub ahead of print date: 5 June 2013
Published date: 1 October 2013
Keywords:
davis–januszkiewicz space, moment–angle complex, polyhedral product, shifted complex, homotopy type
Organisations:
Pure Mathematics
Identifiers
Local EPrints ID: 356515
URI: http://eprints.soton.ac.uk/id/eprint/356515
ISSN: 0001-8708
PURE UUID: fe660993-b91f-41de-9f82-db69dd1b5bb7
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Date deposited: 13 Sep 2013 16:19
Last modified: 15 Mar 2024 03:45
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