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，对于素数$3$-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致叶状结构（$CTF$）。 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）的斜率集合来表述。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$NLS$的情况。 我们很多技术性工作在于证明在$\partial M$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，且这些区间的端点是有理数；特别是，它在斜率空间中是闭的。 这涉及到将 Tao Li 关于层状分支曲面的结果推广到带有边界的流形设置中。 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子集，我们称之为例外的$CTF$检测到的斜率。 这个集合包括与之相关的德恩填充不接受可共向紧致叶状结构的$CTF$检测到的斜率。 我们认为这个例外集合对于理解非常重要。 在本文中，我们证明了例外斜率的集合是有限的。 然而，许多问题仍然开放。 最后，在文章的最后一节，我们提供了该领域之前工作的结构化综述。 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.