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

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.