Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory
Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory
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族 Seiberg-Witten 理论对 4 流形上群作用的平滑阻碍
DOI：
10.1016/j.aim.2019.106730
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发表时间：
2018
影响因子：
1.7
通讯作者：
David Baraglia
中科院分区：
文献类型：
--
作者：
David Baraglia
设X是光滑的、紧的、定向的四维流形。在Li-Liu，Ruberman，中村和Konno的工作基础上，我们考虑了Seiberg-Witten理论的一个族版本，并利用群同态得到了X上某些群作用存在的障碍.这些障碍表明，H2（X，Z）上的某些保持交形式的群作用不能通过双同态提升为同一群在X上的作用.使用我们的障碍，我们构建了许多例子的群体行动，可以实现连续，但不能实现任何可微结构顺利。例如，我们构造了紧致单连通4-流形X和对合f：H2（X，Z）→ H2（X，Z），使得f可以由X上的连续对合或对合同态实现，但不能由X上的任何光滑结构的对合对合同态实现。
Let X be a smooth, compact, oriented 4-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on X by diffeomorphisms. The obstructions show that certain group actions on H 2 (X, Z) preserving the intersection form can not be lifted to an action of the same group on X by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differentiable structure. For example, we construct compact simply-connected 4-manifolds X and involutions f: H 2 (X, Z)→ H 2 (X, Z) such that f can be realised by a continuous involution on X or by a diffeomorphism, but not by an involutive diffeomorphism for any smooth structure on X.