Examples suggest that there is a correspondence between L-spaces and 3manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the 2-fold branched covers of non-split alternating links. To do this we prove that the fundamental group of the 2-fold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in Homeo+(S 1 ).While the trivial group obviously satisfies this criterion, in this paper we will adopt the convention that it is not left-orderable.The left-orderability of 3-manifold groups has been studied in work of Boyer, Rolfsen and Wiest [3]. An argument of Howie and Short [18, Lemma 2] shows that the fundamental group of an irreducible 3-manifold M with positive first Betti number is locally indicable, hence left-orderable [4]. More generally, such a group is left-orderable if it admits an epimorphism to a left-orderable group [3, Theorem 1.1(1)].The aim of this note is to establish a connection between L-spaces and the left-orderability of their fundamental groups. Given the results we obtain and those obtained elsewhere [6,7,8,40], we formalise a question which has received attention in the recent literature in the following conjecture. Conjecture 3. An irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.It has been asked by Ozsváth and Szabó whether L-spaces can be characterized as those closed, connected 3-manifolds admitting no co-orientable taut foliations. Thus, in the context of Conjecture 3 it is interesting to consider the following open questions: Does the existence of a co-orientable taut foliation on an irreducible rational homology 3-sphere imply the manifold has a left-orderable fundamental group? Are the two conditions equivalent? Calegari and Dunfield have shown that the existence of a co-orientable taut foliation on an irreducible atoroidal rational homology 3-sphere Y implies that π 1 (Y ) has a left-orderable finite index subgroup [5, Corollary 7.6]. Of course, an affirmative answer to Conjecture 3, combined with [31, Theorem 1.4], would prove that the existence of a co-orientable taut foliation implies left-orderable fundamental group.Our first result verifies the conjecture in the case of Seifert fibred spaces.Theorem 4. Suppose Y is a closed, connected, Seifert fibred 3-manifold. Then Y is an L-space if and only if π 1 (Y ) is not left-orderable.The proof of this theorem in the case where the base orbifold of Y is orientable depends on results of Boyer, Rolfsen ...

This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M is such a manifold, we show that the type D structure CFD(M ) may be viewed as a set of immersed curves decorated with local systems in ∂M . These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of HF decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of HF . In particular, it follows that a prime rational homology sphere Y with HF (Y ) < 5 must be geometric. Other results include a new proof of Eftekhary's theorem that L-space homology spheres are atoroidal; a complete characterisation of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S 3 , we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.