论文标题
一维叶子的叶片空间的同型叶子在非紧凑型表面上的叶片
Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves
论文作者
论文摘要
令$ z $为一个非紧密的二维流形，它是从一个开放式带$ \ mathbb {r} \ times（0,1）$的一家族获得边界间隔的$ \ mathbb {r} \ times（0,1）$的。每个这样的条带都有一个自然的叶子，将平行线$ \ mathbb {r} \ times t $，$ t \ in（0,1）$和边界间隔提供所有$ z $的边界间隔。用$ \ Mathcal {h}（z，δ）$表示$ z $的所有同构的组组，将$Δ$映射到叶子上，以及$ \ mathcal {h}（z/δ）$ en of相同的compact的叶子空间的同型同态同构的同型同态的同构态度。最近，作者用一组特定图$ g $的自动形态识别出同型组$π_0\ mathcal {h}（z，δ）$，并带有编码从strips中粘合$ z $的组合的附加结构。该图在某种意义上是叶子$ z/δ$的双重偶。 另一方面，对于\ Mathcal {h}（z，δ）$ heaus诱导的置换$ k $的$δ$的$ k $实际上是$ z/δ$的同构，$Δ$的叶子$ h \ apsto k $的同构是一种同构的， $ψ：\ Mathcal {h}（δ）\ to \ Mathcal {h}（z/δ）$。本文的目的是表明$ψ$诱导相应的同型组$ψ_0：π_0\ Mathcal {h}（z，Δ）\toπ_0\toπ_0\toπ_0\ mathcal {h}（z/δ）$，结果是Injextive或Injextive co n nm MathEl $ \ kernel $ {这给出了$π_0\ mathcal {h}（z，δ）$的双重描述。
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.
