Homotopy of gauge groups over non-simply-connected five-dimensional manifolds
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Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper, we combine them together to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to Bott periodicity. Our treatments here are also very effective for rational gauge groups in general context, and applicable for higher dimensional manifolds.
Keywordsgauge group 5-manifolds homotopy decompositions homotopy exponents homotopy groups rational homotopy theory Bott periodicity
MSC(2010)57S05 55P15 55P40 54C35 55P62 57R19 58B05 81T13
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This work was supported by Postdoctoral International Exchange Program for Incoming Postdoctoral Students under Chinese Postdoctoral Council and Chinese Postdoctoral Science Foundation, Chinese Postdoctoral Science Foundation (Grant No. 2018M631605) and National Natural Science Foundation of China (Grant No. 11801544). The author thanks Haibao Duan for bringing the non-simply connected 5-manifolds to his attention. He is indebted to Daisuke Kishimoto for suggestions about adding more applications of the decompositions on an early version of this paper. He also thanks Fred Cohen, and the anonymous referees most warmly for their careful reading of the manuscript and many helpful suggestions which have improved the paper.
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