Homotopy theory of gauge groups over certain 7-manifolds

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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Identifiers

ORCID for Stephen Theriault: orcid.org/0000-0002-7729-5527

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