Counting homotopy types of certain gauge groups

Abstract

Two original studies into the homotopy theory of gauge groups are presented. In the first the number of homotopy types of Sp(3)-gauge groups over S4 are counted, obtaining exact odd primary information and best possible 2-local bounds. This method follows a thorough examination of the commutator product Sp(1) ^ Sp(3) ! Sp(3) to draw its conclusions. In the second work the homotopy types of U(n)-gauge groups over S4 and CP2 are examined. Homotopy decompositions of the U(n)-gauge groups over S4 are given in terms of certain SU(n) and PU(n)-gauge groups and these are then use these to count the number of U(2)-, U(3)- and U(5)-gauge groups over S4. Over CP2 the problem is delicate. Numerous results for the U(n)-gauge groups are given for general values of n, including certain homotopy decompositions and p-local properties. In the final section the previous results are applied to the case of U(2) to obtain the most complete statements. The final part of the thesis is a survey article detailing the history of the homotopy theory of gauge groups. Inuential papers and turning points in the subject are discuss, and the survey ends with an outlook on possible future directions for research.

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

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