3维流形的伪同胚
Pseudo-isotopies of 3-manifolds with infinite fundamental groups
AI总结：
本文研究了3维流形的伪同胚群及其同胚群的性质，证明了其无限秩的交换群结构。
摘要：
假定Y是一个紧连通定向的3维流形，可能带有边界，且π₁(Y)是无限的。令Diff_∂(I×Y)表示I×Y的自同胚群，其在边界附近等于恒等映射。令Diff_{PI}(I×Y)表示Diff_∂(I×Y)的子群，其中元素伪同胚于恒等映射。对同胚定义Homeo_∂(I×Y)、Homeo_{PI}(I×Y)类似。我们证明了自然映射π₀Diff_{PI}(I×Y)→π₀Homeo_{PI}(I×Y)的秩为无限。作为结果，π₀Diff_{PI}(I×Y)、π₀Diff_∂(I×Y)、π₀Homeo_{PI}(I×Y)、π₀Homeo_∂(I×Y)都是无限秩的交换群。我们还证明π₀C(Y)包含一个无限秩的交换子群，且π₀C(I×Y)映射到一个无限秩的交换群。这些结果通过研究barbell同胚在嵌入弧空间和配置空间上的作用来证明。
英文摘要：
Suppose $Y$ is a compact, connected, oriented 3-manifold possibly with boundary, such that $π_1(Y)$ is infinite. Let $\operatorname{Diff}_\partial(I\times Y)$ denote the group of self-diffeomorphisms of $I\times Y$ that are equal to the identity near the boundary. Let $\operatorname{Diff}_{PI}(I\times Y)$ denote the subgroup of $\operatorname{Diff}_\partial(I\times Y)$ consisting of elements pseudo-isotopic to the identity. Define $\operatorname{Homeo}_\partial(I\times Y)$, $\operatorname{Homeo}_{PI}(I\times Y)$ similarly for homeomorphisms. We show that the canonical map $π_0\operatorname{Diff}_{PI}(I\times Y) \to π_0\operatorname{Homeo}_{PI}(I\times Y)$ is of infinite rank. As a consequence, $π_0\operatorname{Diff}_{PI}(I\times Y)$, $π_0\operatorname{Diff}_{\partial}(I\times Y)$, $π_0\operatorname{Homeo}_{PI}(I\times Y)$, $π_0\operatorname{Homeo}_{\partial}(I\times Y)$ are all abelian groups of infinite rank. We also prove that $π_0\,C(Y)$ contains an abelian subgroup of infinite rank, and $π_0\,C(I\times Y)$ admits a surjection to an abelian group of infinite rank, where $C(X)$ denotes the concordance automorphism group $\operatorname{Diff}(I\times X, \{0\}\times X\cup I\times \partial X)$ or $\operatorname{Homeo}(I\times X, \{0\}\times X\cup I\times \partial X)$. These results are proved by studying the actions of barbell diffeomorphisms on the spaces of embedded arcs and configuration spaces.
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