$\widetilde{H}$-配边、无限循环覆盖与实Seiberg--Witten理论
$\widetilde{H}$-cobordisms, infinite cyclic covers, and real Seiberg--Witten theory
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中文总结 AI 辅助
本文通过实Seiberg--Witten理论构造渐近不变量，区分光滑与拓扑$\widetilde{H}$-配边，证明核含$\mathbb{Z}$子群，自旋版本含$\mathbb{Z}^\infty$子群。
中文摘要 AI 辅助
我们研究了由Kawauchi在1976年利用无限循环覆盖引入的区分同调柄的$\widetilde{H}$-配边。尽管自Kawauchi的工作以来，规范理论和Floer理论不变量得到了广泛发展，但此前没有已知的不变量能够区分光滑与拓扑$\widetilde{H}$-配边。在本文中，我们通过将实Seiberg--Witten理论应用于有限循环覆盖，构造了区分同调柄的渐近不变量。利用这些不变量，我们证明了从光滑$\widetilde{H}$-配边群到其拓扑对应物的自然映射的核包含一个同构于$\mathbb{Z}$的子群。我们还定义了这些群的自旋版本，并证明了相应自然映射的核包含一个同构于$\mathbb{Z}^\infty$的子群。
英文摘要
We study $\widetilde{H}$-cobordisms of distinguished homology handles, introduced by Kawauchi in 1976 using infinite cyclic covers. Despite the extensive development of gauge-theoretic and Floer-theoretic invariants since Kawauchi's work, none were previously known to distinguish smooth and topological $\widetilde{H}$-cobordism. In this paper, we construct asymptotic invariants of distinguished homology handles by applying real Seiberg--Witten theory to finite cyclic covers. Using these invariants, we show that the kernel of the natural map from the smooth $\widetilde{H}$-cobordism group to its topological counterpart contains a subgroup isomorphic to $\mathbb{Z}$. We also define spin versions of these groups and show that the kernel of the corresponding natural map contains a subgroup isomorphic to $\mathbb{Z}^\infty$.
发表机构
- University of Cambridge（剑桥大学）
- Seoul National University（首尔大学）
- Kyoto University（京都大学）
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