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Homotopy of gauge groups over high-dimensional manifolds

Part of: Differential topology Homotopy theory Maps and general types of spaces defined by maps Fiber spaces and bundles

Published online by Cambridge University Press:  05 February 2021

Ruizhi Huang
Ruizhi Huang*
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, China (huangrz@amss.ac.cn)
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Abstract

The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high-dimensional manifolds. To be more specific, we study gauge groups of bundles over (n − 1)-connected closed 2n-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of 2n-manifolds are also considered. In all the cases, we show various homotopy decompositions of gauge groups. The methods are combinations of manifold topology and various techniques in homotopy theory.

Keywords

Gauge groupsHomotopy suspensionsHomotopy decompositionsHomotopy exponents

MSC classification

Primary: 55P15: Classification of homotopy type 55P40: Suspensions 54C35: Function spaces
Secondary: 55R25: Sphere bundles and vector bundles 57R19: Algebraic topology on manifolds 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 182 - 208
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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