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.

中文翻译：

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高维流形上规范群的同伦

近几十年来，规范群的同伦理论受到了相当多的关注。在这项工作中，我们研究了一些高维流形上规范群的同伦理论。更具体地说，我们在 (n− 1)-连接闭合 2n-歧管，其分类由 Wall 和 Freedman 在组合类别中确定。我们还根据 James 和 Whitehead 的经典著作研究了球丛总流形的规范群。此外，其他类型的 2n- 歧管也被考虑在内。在所有情况下，我们展示了规范群的各种同伦分解。这些方法是流形拓扑和同伦理论中各种技术的结合。