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

September 2017

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

Text

** Ingrid MembrilloThesis
- Version of Record**
## More information

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

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Date deposited: 05 Oct 2018 11:41

Last modified: 26 Oct 2023 01:51

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## Contributors

Author:
Ingrid Amaranta Membrillo Solis

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