Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds.

中文翻译：

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扭曲的唐纳森不变量

流形的基本群对其底层的光滑结构有很深的影响。在本文中，我们介绍了杨-米尔斯规范理论中唐纳森不变量的一个新变体，在基本群是自由阿贝尔的情况下，由 4 流形的 Picard 群扭曲。然后我们通过使用非交换几何的框架将其推广到基本群的一般情况。我们还验证了我们的不变量区分了一些同胚 4 流形之间的平滑结构。