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arXiv 2202.07770math.GTmath.ATmath.DSmath.GN
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具有非紧叶的非紧曲面上一维叶理的叶空间的同伦群

Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Sergiy Maksymenko, Eugene Polulyakh

备注：
AMSart, 15 pages, 1 figure
发表信息：
Proceedings of the International Geometry Center, vol. 14, no. 4 (2021), 271-290
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摘要：

设$Z$是一个非紧的二维流形，它是由一族具有边界区间的开带$\mathbb{R}\times(0,1)$通过沿其边界区间对胶合这些开带而得到的.每一个这样的条带都有一个自然的叶理，分为平行线$\mathbb{R}\times t$、$t\in(0,1)$和边界间隔，边界间隔在所有$Z$上给出一个叶理$Δ$。用$\mathcal{H}(Z,Δ)$表示将$Δ$的叶映射到叶上的$Z$的所有同胚群，用$\mathcal{H}(Z/Δ)$表示赋予相应紧开拓扑的叶空间的同胚群。最近，作者确定了同伦群$π_0\mathcal{H}(Z,Δ)$与一组自同构的某个图$G$与额外的结构，编码的组合胶合$Z$从条。该图在某种意义上是叶$Z/Δ$空间的对偶。另一方面，对于每个$h\in\mathcal{H}(Z,Δ)$，$Δ$的叶的导出置换$k$实际上是$Z/Δ$的同胚，并且对应的$h\mapsto k$是同态$ψ:\mathcal{H}(Δ)\to\mathcal{H}(Z/Δ)$。本文的目的是表明$ψ$诱导相应的同伦群$ψ_0:π_0\mathcal{H}(Z,Δ)\toπ_0\mathcal{H}(Z/Δ)$的同态原来是内射或有一个核心$\mathbb{Z}_2$。这给出了$π_0\mathcal{H}(Z,Δ)$在叶空间方面的对偶描述。

英文摘要：

Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals which gives a foliation $Δ$ on all of $Z$. Denote by $\mathcal{H}(Z,Δ)$ the group of all homeomorphisms of $Z$ that maps leaves of $Δ$ onto leaves and by $\mathcal{H}(Z/Δ)$ the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group $π_0\mathcal{H}(Z,Δ)$ with a group of automorphisms of a certain graph $G$ with the additional structure which encodes the combinatorics of gluing $Z$ from strips. That graph is in a certain sense dual to the space of leaves $Z/Δ$. On the other hand, for every $h\in\mathcal{H}(Z,Δ)$ the induced permutation $k$ of leaves of $Δ$ is in fact a homeomorphism of $Z/Δ$ and the correspondence $h\mapsto k$ is a homomorphism $ψ:\mathcal{H}(Δ)\to\mathcal{H}(Z/Δ)$. The aim of the present paper is to show that $ψ$ induces a homomorphism of the corresponding homeotopy groups $ψ_0:π_0\mathcal{H}(Z,Δ)\toπ_0\mathcal{H}(Z/Δ)$ which turns out to be either injective or having a kernel $\mathbb{Z}_2$. This gives a dual description of $π_0\mathcal{H}(Z,Δ)$ in terms of the space of leaves.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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