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Symmetries in toric topology

Symmetries in toric topology
Symmetries in toric topology
A polyhedral product (X,A)K is determined by a finite simplicial complex K and m pairs of topological spaces. The functorial property of polyhedral products implies two types of symmetries of polyhedral products induced by symmetries of simplicial complexes and by group actions on the topological pairs. In this thesis, we consider these two types of symmetries.

In the case of G-polyhedral products induced by a simplicial G-complex K, we show that the homotopy decomposition Sigma (X,A)K due to Bahri-Bendersky-Cohen-Gitler [3] is homotopy G-equivariant after suspension. It implies a homological decomposition of Hi((X,A)K) in terms of G-representations, which we rely on to study the representation stability of polyhedral product in the sense of Church-Farb [15].

The torus actions on moment-angle complexes ZK is a special case of actions on polyhedral products induced by actions on the topological pairs. In this thesis, we compute the cohomology of the quotient ZK/S1 under free circle actions and introduce a chain complex (C*(L),δ) whose homology isomorphic to H*(ZK/S1;R). For certain cases K, we determine the homotopy types of ZK/S1.
University of Southampton
Fu, Xin
1190b059-0a15-4312-8321-86b04f82fd36
Fu, Xin
1190b059-0a15-4312-8321-86b04f82fd36
Grbic, Jelena
daaea124-d4cc-4818-803a-2b0cb4362175

Fu, Xin (2019) Symmetries in toric topology. University of Southampton, Doctoral Thesis, 124pp.

Record type: Thesis (Doctoral)

Abstract

A polyhedral product (X,A)K is determined by a finite simplicial complex K and m pairs of topological spaces. The functorial property of polyhedral products implies two types of symmetries of polyhedral products induced by symmetries of simplicial complexes and by group actions on the topological pairs. In this thesis, we consider these two types of symmetries.

In the case of G-polyhedral products induced by a simplicial G-complex K, we show that the homotopy decomposition Sigma (X,A)K due to Bahri-Bendersky-Cohen-Gitler [3] is homotopy G-equivariant after suspension. It implies a homological decomposition of Hi((X,A)K) in terms of G-representations, which we rely on to study the representation stability of polyhedral product in the sense of Church-Farb [15].

The torus actions on moment-angle complexes ZK is a special case of actions on polyhedral products induced by actions on the topological pairs. In this thesis, we compute the cohomology of the quotient ZK/S1 under free circle actions and introduce a chain complex (C*(L),δ) whose homology isomorphic to H*(ZK/S1;R). For certain cases K, we determine the homotopy types of ZK/S1.

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Published date: July 2019

Identifiers

Local EPrints ID: 436387
URI: http://eprints.soton.ac.uk/id/eprint/436387
PURE UUID: 2778d481-c2d9-411e-b372-51a563182632
ORCID for Jelena Grbic: ORCID iD orcid.org/0000-0002-7164-540X

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Date deposited: 09 Dec 2019 17:31
Last modified: 10 Dec 2019 01:36

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Contributors

Author: Xin Fu
Thesis advisor: Jelena Grbic ORCID iD

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