标题： 斜率检测，紧致叶状结构，以及相对L空间猜想 显示英文标题

标题： Slope detection, taut foliations, and the relative L-space conjecture

摘要： $L$-空间猜想断言，对于素的$3$-流形，三个性质是等价的：不是$L$-空间（$NLS$），其基本群是左序的（$LO$），并且可以接受一个共定向的紧化叶状结构（$CTF$）。 在本文中，我们引入了针对纽结流形 $M$的 $L$-空间猜想的相对版本，该版本是通过在 $\partial M$上由Heegaard Floer同调、左序和叶状结构分别表征（即检测）的斜率集来表述的。我们给出了斜率检测的统一表征，并猜想相对的 $L$-空间猜想与环面流形的 $L$-空间猜想等价。我们确认了这一等价性对于性质 $CTF$和 $NLS$。 我们的许多技术工作在于证明在$\partial M$上由$CTF$检测到的斜率集合是有限个可能退化的闭区间的并集，且端点是有理数；特别是，它在斜率空间中是闭的。 这涉及到将 Tao Li 关于层状分支表面的结果推广到带有边界的流形情形。 在由可共向紧致叶状结构检测到的斜率中，我们识别出一个特殊的子集，我们称之为异常的$CTF$检测到的斜率。 这个集合包括与 Dehn 粘贴不承认可共向紧致叶状结构相关的$CTF$检测到的斜率。 我们认为这个异常集合对于理解相关问题很重要。 在本文中，我们证明了异常斜率的集合是有限的。 然而，仍有许多问题尚未解决。 最后，在文章的最后一节，我们提供了该领域先前工作的结构化综合。 显示英文摘要

摘要： The $L$-space conjecture asserts the equivalence, for prime $3$-manifolds, of three properties: not being an $L$-space ($NLS$), having a left-orderable fundamental group ($LO$), and admitting a co-orientable taut foliation ($CTF$). In this paper we introduce a relative version of the $L$-space conjecture for knot manifolds $M$, stated in terms of sets of slopes on $\partial M$ characterised (i.e. detected) by Heegaard Floer homology, left-orders, and foliations, respectively. We give a unified characterisation of slope detection, and conjecture that the relative $L$-space conjecture is equivalent to the $L$-space conjecture for toroidal manifolds. We confirm this equivalence for the properties $CTF$ and $NLS$. Much of our technical work lies in proving that the set of $CTF$-detected slopes on $\partial M$ is a finite union of possibly degenerate closed intervals with rational endpoints; in particular, it is closed in the space of slopes. This involves generalizing results of Tao Li on laminar branched surfaces to the setting of manifolds with boundary. Within the slopes detected by co-orientable taut foliations, we identify a special subset, which we call exceptional $CTF$-detected slopes. This set includes $CTF$-detected slopes whose associated Dehn fillings don't admit co-orientable taut foliations. We believe this exceptional set is important to understand. In this article, we show that the set of exceptional slopes is finite. However, many questions remain open. Finally, in the last section of the article we provide a structured synthesis of previous work in the area.