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Homotopy theory of gauge groups over certain 7-manifolds

Homotopy theory of gauge groups over certain 7-manifolds
Homotopy theory of gauge groups over certain 7-manifolds
The gauge groups of principal G-bundles over low dimensional spaces have been extensively studied in homotopy theory due to their connections to other areas in mathematics, such as the Yang-Mills gauge theory in mathematical physics. In 2011 Donaldson and Segal established the mathematical set-up to construct new gauge theories over high dimensional spaces.

In this thesis we study the homotopy theory of gauge groups over 7-manifolds that arise as total spaces of S 3 -bundles over S 4 and their connected sums. We classify principal G-bundles over manifolds M up to isomorphism in the following cases:

(1) M is an S 3 -bundle over S 4 with torsion-free homology;

(2) M is an S 3 -bundle over S 4 with non-torsion-free homology and π6(G) = 0;

(3) M is a connected sum of S 3 -bundles over S 4 with torsion-free homology and π6(G) = 0.

We obtain integral homotopy decomposition of the gauge groups in the cases for which the manifold is either a product of spheres, or a twisted product of spheres, or a connected sum of those. We obtain p-local homotopy decompositions of the loop spaces of the gauge groups in the cases for which the manifold has torsion in homology. Gauge groups of principal G-bundles over manifolds homotopy equivalent to S 7 are classified up to a p-local homotopy equivalence
University of Southampton
Membrillo Solis, Ingrid Amaranta
94b16293-285b-4bf0-a1e1-b590c6b8b50c
Membrillo Solis, Ingrid Amaranta
94b16293-285b-4bf0-a1e1-b590c6b8b50c
Theriault, Stephen
5e442ce4-8941-41b3-95f1-5e7562fdef80

Membrillo Solis, Ingrid Amaranta (2017) Homotopy theory of gauge groups over certain 7-manifolds. University of Southampton, Doctoral Thesis, 129pp.

Record type: Thesis (Doctoral)

Abstract

The gauge groups of principal G-bundles over low dimensional spaces have been extensively studied in homotopy theory due to their connections to other areas in mathematics, such as the Yang-Mills gauge theory in mathematical physics. In 2011 Donaldson and Segal established the mathematical set-up to construct new gauge theories over high dimensional spaces.

In this thesis we study the homotopy theory of gauge groups over 7-manifolds that arise as total spaces of S 3 -bundles over S 4 and their connected sums. We classify principal G-bundles over manifolds M up to isomorphism in the following cases:

(1) M is an S 3 -bundle over S 4 with torsion-free homology;

(2) M is an S 3 -bundle over S 4 with non-torsion-free homology and π6(G) = 0;

(3) M is a connected sum of S 3 -bundles over S 4 with torsion-free homology and π6(G) = 0.

We obtain integral homotopy decomposition of the gauge groups in the cases for which the manifold is either a product of spheres, or a twisted product of spheres, or a connected sum of those. We obtain p-local homotopy decompositions of the loop spaces of the gauge groups in the cases for which the manifold has torsion in homology. Gauge groups of principal G-bundles over manifolds homotopy equivalent to S 7 are classified up to a p-local homotopy equivalence

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Published date: September 2017

Identifiers

Local EPrints ID: 424732
URI: http://eprints.soton.ac.uk/id/eprint/424732
PURE UUID: 6245ce17-3c4c-4a7c-aaec-592fc1d66fcc
ORCID for Stephen Theriault: ORCID iD orcid.org/0000-0002-7729-5527

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Date deposited: 05 Oct 2018 11:41
Last modified: 14 Mar 2019 01:35

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