Homotopy types of gauge groups of principal bundles with certain non-simply connected structure groups

Abstract

The gauge group of a principal G-bundle P over a space X is the group of G-equivariant homeomorphisms of P that cover the identity on X. To date, the study of the homotopy theory of gauge groups has been focused primarily on principal bundles whose structure groups are simply-connected, mainly due to the inherent complexity of the case of non-simply-connected structure groups.
In this thesis, we carry out a systematic study of the homotopy types of gauge groups of principal bundles with two families of non-simply connected structure groups: namely, the projective unitary groups PU(n), particularly with n prime, and the complex spin groups Spinc(n). These are defined as quotients of U(n) by its centre, and of the product Spin(n) U(1) by the diagonal action, respectively.
We examine the relation between the gauge groups of SU(n)- and PU(n)-bundles over the even dimensional sphere S2i, with 2 i n. As special cases, for U(5)-bundles over S4, we show that there is a rational or p-local equivalence G2;k '(p) G2;l for any prime p if, and only if, (120; k) = (120; l), while for PU(3)-bundles over S6 there is an integral equivalence G3;k ' G3;l if, and only if, (120; k) = (120; l).
We also study the gauge groups of bundles over S4 with Spinc(n) as structure group and show that there is a decomposition Gk(Spinc(n)) ' S1 Gk(Spin(n)). This implies that the homotopy theory of Spinc(n)-gauge groups reduces to that of Spin(n)-gauge groups over S4. We then advance on what is known by providing a partial classification for Spin(7)- and Spin(8)-gauge groups over S4.

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Identifiers

ORCID for Simon Rea: orcid.org/0000-0002-6822-0523

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

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