Floer simple manifolds and L-space intervals (Q1678155)

An L-space is a closed \(3\)-manifold with simplest Heegaard Floer homology. More precisely, a rational homology \(3\)-sphere \(M\) is called an L-space if its Heegaard Floer homology \(\widehat{HF}(M,\mathfrak{s})\) is isomorphic to \(\mathbb{Z}\) for each Spin\(^c\) structure \(\mathfrak{s}\) on \(M\). In the paper under review, the set of L-space fillings for a compact connected oriented \(3\)-manifold with torus boundary is examined. Let \(Y\) be such a manifold, and let \(\mathcal{L}(Y)\) be the set of L-space filling slopes. This set may be empty, but the authors claim that \(Y\) admits at least two L-space fillings if and only if \(Y\) admits a Dehn filling whose core knot is a Floer simple knot. The latter condition means that the core knot \(K\) of the Dehn filling \(Y(\alpha)\) has knot Floer homology \(\widehat{HFK}(K)=\mathbb{Z}^{|H_1(Y(\alpha))|}\). If \(Y\) satisfies the latter condition, then \(Y\) is said to be Floer simple. The first main result shows that if \(Y\) is Floer simple, then \(\mathcal{L}(Y)\) is determined by the Turaev torsion \(\tau(Y)\) and a single Floer simple filling slope. The second main result clarifies when the splice of two rational homology solid tori is an L-space under some extra condition. This is a step towards proving a conjecture by \textit{S. Boyer} and \textit{A. Clay} [Adv. Math. 310, 159--234 (2017; Zbl 1381.57003)] and \textit{J. Hanselman} [Quantum Topol. 8, No. 4, 715--748 (2017; Zbl 1386.57017)]. As an application, a new direct proof of the known fact that a Seifert fibered space over the sphere is an L-space if and only if it does not admit a coorientable taut foliation is given.