Floer Simple Manifolds and L-Space Intervals

An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call"Floer simple,"we construct an invariant which computes the interval of L-space filling slopes from the Turaev torsion and a given slope from the interval's interior. As applications, we give a new proof of the classification of Seifert fibered L-spaces over $S^2$, and prove a special case of a conjecture of Boyer and Clay about L-spaces formed by gluing three-manifolds along a torus.


Introduction
An oriented rational homology 3-sphere Y is called an L-space if the Heegaard Floer homology HF (Y ) satisfies HF (Y, s) ≃ Z for each Spin c structure s on Y . Recent interest in the topological meaning of this condition has been stirred by a conjecture of Boyer, Gordon, and Watson [7], which states that a prime oriented three-manifold Y is an L-space if and only if π 1 (Y ) is non left-orderable. Subsequently, Boyer and Clay [6] studied a relative version of this problem for manifolds with toroidal boundary.
In this paper, we study the set of L-space fillings of a connected manifold Y with a single torus boundary component. If Y is such a manifold, we let Sl(Y ) = {α ∈ H 1 (∂Y ) | α is primitive}/ ± 1 be the set of slopes on ∂Y . Sl(Y ) is a one-dimensional projective space defined over the rational numbers. If we fix a basis µ, λ for H 1 (Y ), we can identify Sl(Y ) with Q := Q∪{∞} via the map aµ + bλ → a/b. We denote by Y (α) the closed manifold obtained by Dehn filling Y with slope α, and let K α ⊂ Y (α) be the core of the filling solid torus. For the set L(Y ) to be nonempty, we must have b 1 (Y ) = 1, which implies that Y is a rational homology S 1 × D 2 . In this paper, we will restrict our attention to manifolds with multiple L-space fillings: that is, for which |L(Y )| > 1. Such manifolds can be easily characterized in terms of their Floer homology. Recall that a knot K in a rational homology sphere Y is Floer simple [20] if the knot Floer homology HF K(K) ≃ Z |H1(Y )| . Equivalently, K is Floer simple if Y is an L-space and the spectral sequence from HF K(K) to HF (Y ) degenerates.
Definition 1.2. A compact oriented three-manifold Y with torus boundary is Floer simple if it has some Dehn filling Y (α) whose core K α is a Floer simple knot in Y (α). If K α ⊂ Y (α) is Floer simple, then the Floer homology of any surgery on K α can be determined from HF K(K α ) using the Ozsváth-Szabó mapping cone. The knot Floer homology, in turn, is determined by the Turaev torsion τ (Y ) via the relation established in Proposition 2.1. It follows that if Y is Floer simple, then the Floer homology of any Dehn filling of Y can be determined from the Turaev torsion together with a single α ∈ L(Y ). In particular, we can determine L(Y ) from this data, as described below.
Write H 1 (Y ) = Z ⊕ T , where T is a torsion group, and let φ : H 1 (Y ) → Z be the projection. Properly normalized, τ (Y ) can be written as a sum where a h = 1 for all but finitely many h ∈ H 1 (Y ) with φ(h) > 0, and a 0 = 0. For example, if H 1 (Y ) = Z, then where the Alexander polynomial ∆(Y ) is normalized to be an element of Z[t] and we expand the denominator as a Laurent series in positive powers of t. and write D τ >0 (Y ) for the subset of D τ (Y ) consisting of those elements with φ(h) > 0. Let [l] ∈ Sl(Y ) be the homological longitude (i.e. l is a primitive element of H 1 (Y ) such that ι(l) is torsion. ) We can now state our first main theorem: Theorem 1.6. If Y is Floer simple, then either D τ >0 (Y ) = ∅ and L(Y ) = Sl(Y ) \ [l], or D τ >0 (Y ) = ∅ and L(Y ) is a closed interval whose endpoints are consecutive elements of ι −1 (D τ >0 (Y )). Given τ (Y ) and a Floer simple filling slope α for Y , it is thus straightforward to determine L(Y ): the torsion determines the set D τ (Y ), and L(Y ) is the smallest interval with endpoints in ι −1 (D τ >0 (Y )) which contains α in its interior. 1.1. Splicing. Theorem 1.6 can be used to address a problem raised by Boyer and Clay in [6]. Suppose that Y 1 and Y 2 are rational homology solid tori, and that ϕ : ∂Y 1 → ∂Y 2 is an orientation reversing diffeomorphism. The manifold Y ϕ = Y 1 ∪ ϕ Y 2 is said to obtained by splicing Y 1 and Y 2 together by ϕ.
In [6], Boyer and Clay studied how the presence of structure ( * ) on Dehn fillings of the pieces Y 1 and Y 2 relates to the presence of structure ( * ) on the splice Y ϕ , where structure ( * ) could be one of three things: 1) a coorientable taut foliation; 2) a left-ordering on π 1 (Y ϕ ); or 3) a nontrivial class in HF red (Y ϕ ) (as HF red vanishes on, and only on, L-spaces). When Y 1 and Y 2 are graph manifolds, they obtained very strong results in cases 1) and 2), in addition to less complete results in the third case. The analogy with the first two cases suggests the following conjecture, which is implicit in the work of Boyer and Clay and stated explicitly in certain cases by Hanselman [16]. Conjecture 1.7. Suppose that Y 1 and Y 2 as above are boundary incompressible, and let L • i be the interior of L(Y i ) ⊂ Sl(Y i ). Then Y ϕ is an L-space if and only if ϕ * (L • 1 )∪L • 2 = Sl(Y 2 ). In particular, the conjecture says that in order for Y ϕ to be an L-space, both Y 1 and Y 2 must be Floer simple. Our second main result is Theorem 1.8. Suppose that Y 1 and Y 2 as above are Floer simple and have D τ = ∅, and that ϕ * (L • 1 ) ∪ L • 2 = ∅. Then Y ϕ is an L-space if and only if ϕ * (L • 1 ) ∪ L • 2 = Sl(Y 2 ). Hanselman and Watson [19] have proved a similar theorem using bordered Floer homology. The restriction that ϕ * (L • 1 ) ∩ L • 2 = ∅ represents a limitation of our approach, rather than anything intrinsic to the problem. To be specific, Theorem 1.8 is proved by writing Y ϕ as surgery on a connected sum of Floer simple knots. When ϕ * (L • 1 ) ∩ L • 2 = ∅, we have no convenient way of representing the splice as surgery on a knot in an L-space. In contrast, Hanselman and Watson's approach does not require this hypothesis, but does need a condition on the bordered Floer homology, which they call simple loop type. In a subsequent joint paper [17], it is shown that the conditions of being Floer simple and being simple loop type are equivalent thus enabling us to remove the hypothesis that ϕ * (L • 1 ) ∩ L • 2 = ∅. The proof of this fact relies on Proposition 3.9 of the current paper, where we explicitly compute the bordered Floer homology CF D(Y, µ, λ) of a Floer simple manifold Y for an appropriate choice of µ, λ ∈ H 1 (∂Y ) parametrizing ∂Y .
We briefly discuss those aspects of Conjecture 1.7 which are not covered by Theorem 1.8 and its generalizations. As stated, the conjecture implies that a Floer simple manifold Y with D τ (Y ) = ∅ is boundary compressible. This is easily seen to be the case when H 1 (Y ) ≃ Z, or more generally, when Y is semi-primitive (c.f. Proposition 1.9 below), but in general we have very little idea how to address this question. (Indeed, this seems like the weakest point of the conjecture. ) The other situation which is not addressed by Theorem 1.8 is the case where one or both of Y 1 and Y 2 is not Floer simple. It seems plausible that bordered Floer homology could be used to prove the conjecture when |L(Y 1 )| = 1 and |L(Y 2 )| > 1, or when |L(Y 1 )| = |L(Y 2 )| = 1. In contrast, the case where one or both of the Y i has no L-space fillings seems considerably more difficult to address with current technology.

1.2.
Floer homology solid tori. The class of Floer simple manifolds with D τ >0 = ∅ is of special interest. If Y is a rational homology S 1 × D 2 , we say that Y is semi-primitive if the torsion subgroup of Y is contained in the image of ι, and that Y has genus 0 if H 2 (Y, ∂Y ) is generated by a surface of genus 0. Proposition 1.9. If Y is semi-primitive, the following conditions are equivalent: (1) Y is Floer simple and D τ >0 (Y ) = ∅. (2) Y is Floer simple and has genus 0.
(3) Y has genus 0 and has an L-space filling.
For example, if K ⊂ S 1 × S 2 has a lens space surgery, then the complement of K satisfies the conditions of the proposition. Such knots have been studied by Berge [3], Gabai [15], Cebanu [10], and Buck, Baker and Leucona [2]. Other examples of such manifolds are discussed in section 7.3.
The conditions of Proposition 1.9 are closely related to Watson's notion of a Floer homology solid torus. Suppose that Y is a rational homology S 1 × D 2 with homological longitude l, and that m ∈ H 1 (∂Y ) satisfies m · l = 1. Proposition 1.11. If Y satisfies the conditions of Proposition 1.9, then it is a Floer homology solid torus.
Manifolds with D τ >0 (Y ) = ∅ play an important role in the notion of NLS detection introduced by Boyer and Clay in [6]. If Y is a rational homology S 1 × D 2 and α ∈ Sl(Y ), α is said to be strongly NLS detected if Y (α) is not an L-space; α is NLS detected if certain splicings of Y with a family of Floer homology solid tori are not L-spaces. (For the precise definition, see section 7.2). By Theorem 1.6, the set of strongly NLS detected slopes is either a single point, an open interval in Sl(Y ), or all of Sl(Y ). By combining Theorem 1.8 with some direct geometric computation, we can show Corollary 1.12. If Y is a rational homology S 1 × D 2 , the set of NLS detected slopes in Sl(Y ) is the closure of the set of strongly NLS detected slopes.
In combination with a result of Boyer, Rolfsen, and Wiest [8], this also implies that a Seifert-fibred space over S 2 has non left-orderable π 1 if and only if it is an L-space. The set of Seifert fibred spaces over S 2 which admit a coorientable taut foliation was explicitly described by Jankins and Neumann [24] and Naimi [33], building on a result of Eisenbud, Hirsch, and Neumann [11].
Any Seifert-fibred space over S 2 can be obtained by Dehn filling a Seifert fibred space over D 2 . It follows easily from work of Ozsváth and Szabó [37] that any Seifert fibred space over D 2 is Floer simple, so we can compute the set of L-space filling slopes using Theorem 1.6. The resulting description of the set of Seifert fibred spaces which are not L-spaces agrees with the Jankins-Neumann set, thus giving a new direct proof of Theorem 1.13.
1.4. Discussion. We conclude with some questions about about Floer simple manifolds and their relation to the conjecture of Boyer, Gordon, and Watson. First, we recall the statement of the conjecture. Conjecture 1.14. [7] If Y is a oriented, closed, prime three-manifold, then Y is an L-space if and only if π 1 (Y ) is non left-orderable.
A potentially more tractable subset of this problem, raised by Boyer and Clay [6] is: The characterization of L(Y ) given in Theorem 1.6 should make it possible to conduct more detailed tests of Conjecture 1.14. Since there is already considerable experimental evidence in support of the conjecture, we should also consider what circumstances might explain a positive answer to Question 1. One possible explanation is that the condition of being Floer simple is correlated with some strong geometrical property, which in turn can be related to orderings of π 1 . Question 2. Is there a geometric characterization of Floer simple manifolds which can be stated without reference to Floer homology?
More generally, we think that Floer simple manifolds are a natural class of manifolds whose geometrical properties should be investigated for their own sake. Some evidence in support of this idea is provided by the frequency of Floer simple manifolds among geometrically simple 3-manifolds (as measured by the SnapPea census). Proposition 1.3 may lead readers familiar with the example of L-space knots in S 3 to suspect that the class of Floer simple manifolds is relatively small, but this is not the case. Of the 59,068 rational homology S 1 × D 2 's in the SnapPy census of manifolds triangulated by at most 9 ideal tetrahedra, nearly 20% have multiple finite fillings, and are thus certifiably Floer simple. Moreover, more than two-thirds of the remaining manifolds have Turaev torsion compatible with their being Floer simple. It seems likely that many of these manifolds are Floer simple as well. (The authors thank Tom Brown for sharing these statistics with them.) For those who like other geometries, we note that every Seifert fibred rational homology S 1 × D 2 is Floer simple.
It would be interesting to know what happens to the density of Floer simple manifolds as the complexity increases. Perhaps the most basic question we could ask along these lines is Question 3. Are there infinitely many irreducible Floer simple manifolds with the same Turaev torsion?
1.5. Organization. The remainder of the paper is organized as follows. In section 2, we review some facts about knot Floer homology and the Ozsváth-Szabó mapping cone. These are used in section 3 to prove Proposition 1.3 and to give a characterization of when a given surgery on a Floer simple knot produces an L-space. In this section, we also explain how to compute the bordered Floer homology of a Floer simple manifold. Theorem 1.6 is proved in In Section 4. In Section 5 we apply Theorem 1.6 to Seifert fibred spaces, thus giving a new proof of Theorem 1.13. The proof of Theorem 1.8 is given in Section 6. Finally, in Section 7, we discuss manifolds with D τ >0 = ∅. Acknowledgements: The authors would like to thank Steve Boyer, Tom Brown, Adam Clay, Tom Gillespie, Jonathan Hanselman, Robert Lipshitz, Saul Schleimer, Faramarz Vafaee, and Liam Watson for helpful conversations. We also thank the organizers of the 9th William Rowan Hamilton conference in Dublin, which helped to get this project started.

Knot Floer homology and the Ozsváth-Szabó mapping cone
In this section, we briefly recall some facts about knot Floer homology [36,43,40] which will be used in what follows. First, let us fix some notation. Throughout this section, we assume that K ⊂ Y is an oriented knot in a rational homology sphere. We let Y = Y \ ν(K) be its complement, and denote by µ ∈ H 1 (∂Y ) the class of its meridian. Furthermore, we let T ⊂ H 1 (Y ) be the torsion subgroup, and denote by φ : where the isomorphism is chosen so that φ(µ) > 0.
2.1. Knot Floer homology. The knot Floer homology HF K(K) is a finitely generated abelian group with an absolute Z/2 grading. It decomposes as a direct sum HF K(K) = ⊕ HF K(K, s), where s runs over the set Spin c (Y, ∂Y ) of relative Spin c structures on (Y, ∂Y ). Spin c (Y, ∂Y ) is an affine copy of H 1 (Y ) (aka H 1 (Y ) torsor); it has a free transitive action of H 1 (Y ). The group HF K(K, s) is trivial for all but finitely many s ∈ Spin c (Y, ∂Y ).
Given s ∈ Spin c (Y, ∂Y ), we consider the formal sum where χ( HF K(K, s)) is defined using the absolute Z/2 grading. We view χ s ( HF K(K)) as an element of the group ring Z[H 1 (Y )]; it is known as the graded Euler characteristic of HF K(K). Clearly .
From now on, we will drop s from the notation and view χ( HF K(K)) as an element of For knots in S 3 , it is well-known that χ( HF K(K)) is the Alexander polynomial of K. More generally, we have Proof. HF K(K) can be identified with the sutured Floer homology SF H(Y, γ µ ) [25], where the suture γ µ consists of two parallel copies of µ. The Euler characteristic of the sutured Floer homology can be described as an appropriately formulated torsion [13]. When ∂Y is toroidal, this torsion can be expressed in terms of the Turaev torsion, as in Lemma 6.3 of [13]. (This lemma was stated for links in S 3 , but the proof carries through unchanged.) , which we shall later sometimes call the "Laurent series group ring." As an element of the Novikov ring, τ (Y ) is well-defined up to multiplication by elements of H 1 (Y ). We shall always normalize so that τ (Y ) has the form τ Note that in general, ∆(Y ) = ∆(Y ); an interesting example to consider is the connected sum If K is a knot in S 3 , it is well known that deg ∆(t) ≤ 2g(K), and ∆(K)| t=1 = 1. The following result is a simultaneous generalization of these two facts.
More generally, it is known that HF K(K) determines both the Thurston norm of Y and whether it is fibred [35,34,26]. Since the knot Floer homology of a Floer simple knot is determined by its Euler characteristic, we have 2.2. Differentials. The knot Floer homology of K can be used to compute the Floer homology of surgeries on K. Before we explain how to do this, we must understand the relation between HF K(K) and HF (Y ).
We begin by discussing Spin c structures. There are maps It is easy to see that this is the same as requiring that i h (s 1 ) = i h (s 2 ), and that the equivalence classes are orbits of Spin c (Y, ∂Y ) under the action of µ.
Let s be an equivalence class in Spin c (Y, ∂Y ). After we choose some auxiliary data (a doubly pointed Heegaard diagram for K), Heegaard Floer homology constructs for us a graded group , which are filtered with respect to the Spin c grading in the following sense: if . These differentials satisfy the relations The Spin c grading provides a natural filtration on the latter two complexes, in the sense that ⊕ k<n CF K(K, s + kµ) is a subcomplex of ( CF K(K, s), d 0 + d v ) and ⊕ k>n CF K(K, s + kµ) is a subcomplex of ( CF K(K, s), d 0 +d h ). These filtrations give rise to spectral sequences whose E 1 term is HF K(K, s). We denote byd v ,d h the induced differentials on the E 1 term, so that e.g. CF K(K, d 0 + d v ) is homotopy equivalent to HF K(K,d v ). (Note that these are not the same as the d 1 differentials in the spectral sequence.) The bent complexes measure the Heegaard Floer homology of large integer surgery on K: H(A K,s ) ≃ HF (Y (N µ + λ), i n (s)) for sufficiently large N and an appropriately chosen Spin c structure i N (s) on the filling.
The existence of the Spin c filtration means there are chain maps given by 2.3. The Ozsváth-Szabó mapping cone. Let λ be a longitude for K, so that µ · λ = 1.
The mapping cone of Ozsváth and Szabó [40] relates the Heegaard Floer homology of the filling Y (λ) to the knot Floer homology of K. We recall its construction here.
This isomorphism is realized by a chain homotopy equivalence (The map on homology induced by j is the canonical isomorphism of [27], although we will not use this fact here.) . We form two chain complexes There is a chain map f λ : is a sum of terms in B K,s and B K,s+λ .) Let X λ (K) be the mapping cone of f λ . In [40] , Ozsváth and Szabó prove We make some remarks on the construction. First, it is easy to see that the complex X λ (K) decomposes as a direct sum of complexes whose underlying groups are of the form The summands are on one to one correspondence with elements of the quotient ). The resulting decomposition on homology corresponds to the decomposition of HF (Y (λ)) by Spin c structures.
Second, if F p is the field of order p, where p is a prime, then we can form the complex X λ (K; F p ) = X λ (K) ⊗ F p . It follows from the universal coefficient theorem that Finally, it is often convenient to work with the homology of the complexes A K,s and B K,s , rather than the complexes themselves. We can do this if we use field coefficients. Specifically, fix a field F p , and let A K,s = H(A K,s ⊗F p ), A(K) = ⊕A K,s , B K,s = H(B K,s ⊗F p ), B(K) = ⊕B K,s . Similarly, let v : A K,s → B K,s be the map induced by π v , and h : A K,s → B K,s+λ be the map induced by j • π h . Finally, let C λ (K; F p ) be the chain complex whose underlying group is A(K) ⊕ B(K), with differential given by dx = v(x) + h(x) for x ∈ A(K), dy = 0 for y ∈ B(K).
Proof. The short exact sequence gives a long exact sequence whose boundary map is given by v + h. An exact sequence over a field splits, so we get the statement of the corollary.
2.4. Splicing and surgery. Suppose Y 1 and Y 2 are rational homology solid tori, and that ϕ : ∂Y 1 → ∂Y 2 is an orientation reversing diffeomorphism. The manifold Y ϕ = Y 1 ∪ ϕ Y 2 is obtained by splicing Y 1 and Y 2 together along ϕ. Choose a slope µ 1 ∈ Sl(∂Y 1 ), and let µ 2 = ϕ * (µ) be its image in Sl(∂Y 2 ). Let Y i = Y i (µ i ) be the corresponding Dehn fillings, and let K i = K µi be their cores.
This is well-known, but an understanding of the proof will be useful in what follows, so we sketch it here.
We make two remarks on the utility of this construction. First, it is quite flexible, in the sense that the choice of any meridian µ 1 ∈ Sl(∂Y 1 ) gives a different way of realizing the spliced manifold as a surgery. This flexibility will be useful to us in what follows.
Second, rational surgery on a knot K ⊂ Y amounts to splicing Y with S 1 × D 2 . Suppose µ, λ is our usual basis for H 1 (∂Y ), and that m, l is the standard basis for H 1 (∂S 1 × D 2 ) (so l = [∂D 2 ]). If we glue ∂Y to ∂(S 1 × D 2 ) in such a way that [∂D 2 ] is identified with α = pµ + qλ ∈ H 1 (∂Y ), then it is easy to see that µ is identified with −qm + p * l, where pp * ≡ 1 mod q. Applying the lemma, we see that Y (α) is obtained by integer surgery on a knot K ′ = K#K −q/p ⊂ Y #L(q, −p * ) = Y #L(q, −p).
The knot K −q/p is the unique knot in L(q, −p) whose complement is S 1 × D 2 . (In the notation of [44], it is the simple knot K(q, −p, 1)). It is Floer simple, with Euler characteristic To use Lemma 2.7 to compute the Floer homology of a splice, we need to know how the knot Floer homology behaves under connected sum.
The isomorphism is well-behaved with respect to Spin c structures, in the sense that It is also respects the differentials, in the sense that CF K( , and similarly for d h .
In [41], Ozsváth and Szabó combined the observations above with their mapping cone for integer surgeries to express the Floer homology of any rational surgery as a mapping cone.

Floer Simple Manifolds
In this section we use Ozsváth and Szabó's mapping cone formula to prove Proposition 1.3 and to derive some basic facts about Floer simple manifolds. For the most part, these are straightforward extensions of results in [39], [44], and [4]. We conclude by explaining how to compute the bordered Floer homology of a Floer simple manifold Y in terms of τ (Y ) and a Floer simple filling slope α. Our notation and assumptions are the same as in section 2.
3.1. Proof of Proposition 1.3. Suppose that K ⊂ Y is a knot in an L-space, and that some nontrivial surgery on Y is also an L-space. Note that all the x i 's in the definition must have the same relative Z/2 grading, which is opposite that of the y i 's. Since there are more x i 's than y i 's, the x i contribute to χ( HF K(K)) with positive sign, while the y i 's contribute with negative sign.
Ozsváth and Szabó proved in [39] that if K ⊂ S 3 has an L-space surgery with positive slope, then HF K(K) is a positive chain. The following generalization is an easy consequence of a result of Boileau, Boyer, Cebanu, and Walsh: Lemma 3.2. Suppose that K ⊂ Y is a knot in an L-space, and that some surgery on K is also an L-space. Then HF K(K) consists of coherent chains.
Proof. A surgery on K is positive if the corresponding surgery cobordism is positive definite. Suppose that some positive integral surgery on K is an L-space. By Lemma 6.7 of [4], the bent group A K,s ≃ Z for all s ∈ Spin c (Y, ∂Y ). The proof of Theorem 1.2 of [39] carries over unchanged to show that HF K(K, s) is a positive chain.
Next, suppose that Y ′ is obtained by negative integral surgery on K ⊂ Y , and that Y ′ is an L-space. By reversing the orientation of the surgery cobordism, we see that −Y ′ is obtained by positive surgery on −K ⊂ −Y . −Y ′ is also an L-space, so HF K(−K) consists of positive chains, and HF K(K) consists of negative ones.
Finally, suppose that an L-space Y ′ is obtained by fractional surgery on K. Then Y ′ is obtained by integral surgery on a knot of the form K#K −q/p ⊂ Y #L(q, −p), so HF K(K#K −q/p ) ≃ HF K(K) ⊗ HF K(K −q/p ) is composed of coherent chains. Since K −q/p is Floer simple, it is easy to see that this occurs if and only if HF K(K) is composed of coherent chains. [h].
Proof. We have is a set of coset representatives for the action of µ and The hypothesis that HF K(K) consists of coherent chains implies that the nonzero coefficients of χ( HF K(K, s)) alternate between +1 and −1, and that the outermost coefficients are +1. It follows that the coefficients of the product χ( HF K(K, s))  Proof. By hypothesis, HF K(K) is composed of coherent chains, so to prove that K is Floer simple, it suffices to show that every monomial in χ( HF K(K)) appears with a positive coefficient. As usual, we normalize τ (Y ) = h a h [h] so that a h = 0 whenever φ(h) < 0, and a 0 = 0. We have Both terms in this difference are either 0 or 1. If φ(µ) > φ(h), then a h−µ = 0, while if φ(h) ≥ φ(µ) > Y , then a h = 1 by Proposition 2.2. In either case, we see that the coefficient of [h] in χ( HF K(K)) is either 0 or 1. Proof. Since HF K(K) is composed of positive chains, the homology of each of its bent complexes is Z. Since the homology of the bent complexes computes HF (Y (N µ + λ)) for some N ≫ 0, we see that N µ + λ ∈ L(Y ). Since µ · (N µ + λ) = 1, Proposition 17 of [7] shows that the entire interval [µ, N µ + λ] is contained in L(Y ).
By considering mirrors, we see that if HF K(K) is composed of negative chains, then µ is the right endpoint of a closed interval in L(Y ). It follows that if K is Floer simple, then it is an interior point of an interval in L(Y ). Conversely, if HF K(K) is composed of negative chains but is not Floer simple, then some bent group of K has rank > 1. This implies that Y (N µ + λ) is not an L-space for N ≫ 0. Thus if HF K(K) is composed of coherent chains but is not Floer simple, µ is in not in the interior of L(Y ).
3.2. Surgery on Floer simple knots. We now suppose that K ⊂ Y is Floer simple. We give a graphical criterion for determining whether a given integer surgery on K is an Lspace. To do so, we consider the set In other words, every s ∈ Spin c (Y, ∂Y ) can be written in a unique way as s + nµ, where s ∈ S black and n ∈ Z. We color s black if n = 0, red if n > 0, and blue if n < 0. Now suppose we do surgery along K with slope λ, where µ·λ = 1. We divide Spin c (Y, ∂Y ) into cosets for the action of λ . Each coset L is an affine copy of Z, so it has a natural ordering. Each element of L is colored either black, red, or blue; elements which are sufficiently negative are all colored blue, and elements which are sufficiently positive are all colored red. We say L is properly colored if no red element of L appears before a blue element. Proof. The argument is the same as the proof of Lemma 4.8 in [44]; we sketch it briefly here. We fix a prime p and use the mapping cone to compute HF (Y (λ); F p ). The mapping cone C λ (K) decomposes as a direct sum of chain complexes C L , one for each coset L. Since K is Floer simple, the bent groups A K,s+nλ appearing in one summand are all isomorphic to F p , as are the groups B K,s+nλ . Let h s , v s be the restriction of the maps h, v to A K,s . If s is colored red, the map v s is an isomorphism and h s = 0; if s is colored blue, the map h s is an isomorphism and v s = 0; and if s is colored black, both h s and v s are isomorphisms.
The complex C L takes the form shown in Figure 1, where each colored dot in the top row represents A K,s+nλ ≃ F p , each dot in the bottom row represents B K,s+nλ ≃ F p , and the arrows represent nonzero differentials. The chain of differentials breaks each time we encounter a red or blue dot, thus decomposing C L into smaller summands. Summands corresponding to intervals in L whose endpoints are both red or both blue are acyclic; summands whose left endpoint is blue and whose right endpoint is red have homology in even Z/2 homological degree, and summands whose left endpoint is red and whose right endpoint is blue have homology in odd Z/2 homological degree.
It follows that HF (Y (λ), s) ≃ F p if and only if L is properly colored, and hence that Y (λ) is an F p L-space if and only if every coset is properly colored. Finally, the statement of the proposition follows from the fact that Y (λ) is an L-space if and only it is an F p L-space for every prime p.
3.3. Bordered Floer homology of Floer simple manifolds. In this section, we show that the bordered Floer homology [29] of a Floer simple manifold Y is determined by the Turaev torsion of Y together with a slope in the interior of L(Y ). We very briefly review some facts about bordered Floer homology; for more details see [29,30].
A bordered three-manifold is an oriented three-manifold Y equipped with a parametrization (that is, a minimal handle decomposition) of its boundary. We will restrict our attention to the case where ∂Y = T 2 , in which case a parametrization is specified by a choice of two simple closed curves µ, λ ∈ H 1 (∂Y ) which satisfy µ · λ = 1.
The type D bordered Floer homology CF D(Y, µ, λ) is a differential graded module over a certain F 2 -algebra A(Z) associated to the torus. A(Z) is generated by elements ρ 1 , ρ 2 , ρ 3 , ρ 12 , ρ 23 and ρ 123 corresponding to certain arcs on the boundary of the 0-handle in the handle decomposition of ∂Y , together with a pair of idempotents ι 0 , ι 1 . Following Chapter 11 of [29], we can think of the module structure as being specified by a pair of vector spaces V 0 , V 1 over the field of two elements F 2 , together with linear maps In writing the above, we have assumed that CF D(Y, µ, λ) has been reduced with respect to all provincial differentials, so that where the suture γ µ is two parallel copies of µ, and similarly for γ λ .
Petkova [42] showed that the algebra A(Z) can be given an absolute Z/2 grading, and that CF D(Y, µ, λ) can be given a Z/2 grading compatible with it. Petkova's grading depends on some auxiliary choices, but we can make some statements which are independent of these choices.
Lemma 3.7. The maps D 12 and D 23 preserve the homological Z/2 grading. If D 1 has parity i with respect to the Z/2 grading, then D 2 , D 3 and D 123 have parity 1 + i, i and 1 + i, respectively.
We will also need to know how the D I 's behave with respect to the Spin c grading. Let us write V 0 s := HF K(K µ , s), so we have a decomposition V 0 ≃ ⊕ s V 0 s , and similarly for V 1 , where the indexing sets in the sums are Spin c (Y, γ µ ) and Spin c (Y, γ λ ), as defined in [25]. Elements of Spin c (Y, γ µ ) are represented by homology classes of nonvanishing vector fields on Y with fixed behavior on ∂Y . (Recall that two nonvanishing vector fields are said to be homologous if they are homotopic on the complement of a ball in Y .) The sets Spin c (Y, γ µ ) and Spin c (Y, γ λ ) are in bijection, but not canonically so, since the boundary conditions are different.
This is essentially Lemma 11.42 of [29], but stated so as to clarify the dependence on µ and λ.
Proof. Huang and Ramos [23] have constructed a grading gr on CF D(Y, µ, λ). This grading lives in a set S(H) of homotopy classes of nonvanishing vector fields on Y which satisfy certain boundary conditions. To be specific, for each elementary idempotent ι in the algeba A(Z), there is an associated vector field v ι on ∂Y , and if v ∈ S(H), then v| ∂Y should be equal to v ι for some elementary idempotent ι.
Similarly, Huang and Ramos consider the set G(Z) of homotopy classes of nonvanishing vector fields on ∂Y × [0, 1], subject to the constraint that v| ∂Y ×0 = v ι and v| ∂Y ×1 = v ι ′ for some elementary idempotents ι and ι ′ . They show that G(Z) forms a groupoid under concatenation, and that it acts on the grading set S(H), again by concatenation. In section 2.3 of [23], they construct explicit vector fields v I on ∂Y × [0, 1] associated to each ρ I ; the grading of ρ I x is the vector field v I · gr x, where · denotes the action by concatenation.
The grading of [23] contains the Spin c grading, in the sense that if x is a generator of CF D(Y, µ, λ), then its Spin c grading is s(x) = p(gr x), where p is the forgetful map which takes a homotopy class of vector fields to its homology class. By Theorem 1.3 of [23], if x ∈ CF D(Y, µ, λ), gr ∂x = λ −1 · gr x, where λ is a vector field on ∂Y × [0, 1] which is supported in a ball. It follows that s(∂x) = s(x), and hence that p The fact that G(Z) is a groupoid implies that j is a bijection; j is equivariant with respect to the action of H 1 (Y ) since we can arrange this action to take place in the interior of Y , away from the region in which the concatenation takes place. Similarly, we see that ). The set of homology classes of nonvanishing vector fields on ∂Y × [0, 1] which restrict to v ι0 on one end and v ι1 on the other is an affine copy of The arguments for the other D I 's are very similar. Proposition 3.9. Suppose that Y is Floer simple, that α ∈ Sl(Y ) is a Floer simple filling slope, and that µ, λ ∈ H 1 (∂Y ) satisify µ · λ = 1. Then CF D(Y, µ, λ) is determined by α and τ (Y ).
Proof. It suffices to show that CF D(Y, µ, λ) is determined for one particular choice of µ and λ, since the invariant of any other choice can then be determined using the change of basis bimodules in [30]. We choose µ to be a slope in the interior of L(Y ) such that φ(µ) > Y , and take λ = λ 0 − N µ, where λ 0 is some class with µ · λ 0 = 1, and N ≫ 0. (We will specify below how large N needs to be. ) The knots K µ , K λ are Floer simple, so all the elements of V 0 have the same Z/2 grading. Similarly, all elements of V 1 have the same Z/2 grading. By Lemma 3.7, either D 2 = D 123 = 0 or D 1 = D 3 = 0. To see which of these two options hold, we consider the effect of a Dehn twist along µ. We have where the change of framing bimodule CF DA(τ µ ) is shown in Figure 2.
Denote by D : W 1 → W 1 the contribution to ∂ coming from provincial differentials; then we have H(W 1 , D) = HF K(K µ+λ ). By choosing N sufficiently large, we can ensure that µ + λ = λ 0 − (N − 1)µ is in the interior of L(Y ). It follows that HF K(K µ+λ ) is Floer simple and has dimension equal to Referring to the figure, we see that the only contribution to the provincial differential D comes from the arrow labeled 1 ⊗ ρ 3 . Thus the map ρ 3 : Proof. It is well known [35] that HF K detects the Thurston norm, in the sense that if Choose nonzero elements It follows that j(s max ) must be maximal and j(s min + µ + λ) must be minimal.
We represent CF D(Y, µ, λ) by a directed graph like that shown in Figure 3, with a vertex for each generator and an edge for each potential component of the differential; that is, for each pair of generators x, y whose Z/2 and Spin c gradings are compatible with having D I x = y for some D I , we draw an edge from x to y and label it with D I . Lemma 3.11. Each vertex of the graph associated to CF D(Y, µ, λ) has valence two.
Proof. First suppose that x is a generator of V 0 . We have already seen that D 1 and D 3 are both injective, so x is the starting point of one arrow labeled with D 1 and one arrow labeled with D 3 . D 2 = D 123 = 0, so the only other possible arrows adjacent to x are labeled by D 12 . Now D 12 shifts the Spin c grading by −λ, and φ( then D 12 vanishes for grading reasons. Next, if x is a generator of V 1 , it can be a terminal point of an arrow labeled D 1 or D 3 , and either an initial or a terminal point of a arrow labeled D 23 . We claim that x is a terminal point of an arrow of type D 1 if and only if it is not an initial point of an arrow of type D 23 . To see this, consider s ∈ Spin c (Y, γ µ ). We say s is occupied if s ∈ S[ HF K(K µ )], and unoccupied otherwise; similarly for j(s) ∈ Spin c (Y, γ λ ), but with K λ in place of K µ . The claim is equivalent to saying that if j(s) is occupied, then exactly one of s and j(s) + µ is occupied.
Write j(s) = j(s max ) − α for α ∈ H 1 (Y ). We consider the situation case by case, depending on the value of φ(α).
(2) 0 ≤ φ(α) < φ(µ). In this region, χ( HF K(K µ )) and χ( HF K(K λ )) are both given by given by τ (Y ). By Proposition 2.2, both j(s) and j(s) + µ are always occupied. 4) and is always occupied. Since K µ and K λ are Floer simple, each arrow in the diagram corresponds to a map F 2 → F 2 . To determine the corresponding component of the differential, it suffices to know whether or not this map is 0. We will show that every map corresponding to an arrow in the diagram is nonzero, thus completing the proof of Proposition 3.9. The maps D 1 and D 3 are injective, so any arrow labeled by D 1 or D 3 is nonzero. For the arrows labeled by D 23 , we argue as in the proof of Theorem 11.36 in [29]. By Proposition 11.30 of [29], there are maps D 012 , D 01 , D 0 , D 230 , and D 301 satisfying Since D 2 = D 123 = 0, it follows that if x is not in the image of D 3 , it must be in the image of D 23 , and if x is not in the image of D 1 , D 23 (x) = 0. Comparing with the proof of Lemma 3.11, we see that every arrow in the diagram must correspond to a nonzero map.

Intervals of L-space filling slopes
Now that the "proper coloring" condition of Proposition 3.6 is in place, we are equipped to tackle the problem of describing L-space intervals in terms of D τ (Y ) and a slope from the interior of the L-space interval. We begin by establishing some conventions.

4.1.
Conventions for slopes and homology. If Y is a compact oriented three-manifold with torus boundary, then a slope of Y is a nonseparating, oriented, simple closed curve in ∂Y . Such objects correspond bijectively to primitive elements of H 1 (∂Y )/{±1}, or equivalently, to elements of P(H 1 (∂Y )). Any choice of basis (m, l) for H 1 (∂Y ) specifies homogeneous coordinates nm + n ′ l → [n : n ′ ] on P(H 1 (∂Y )), to which we usually refer in terms of the affinization nm + n ′ l → n/n ′ .
Let ι : H 1 (∂Y ) → H 1 (Y ) be the map induced by inclusion. We fix a basis (m, l) for H 1 (∂Y ) such that l is a generator of ker ι and m · l = 1. The generator l is the homological longitude of Y ; it is well defined up to sign. In contrast, the choice of m is only well defined up to the addition of a multiple of l. Consequently, the numerator of π(nm + n ′ l) = n/n ′ is canonical (up to sign), but the denominator depends on the choice of m.
To Dehn fill Y along a slope µ = nm + n ′ l ∈ H 1 (∂Y ), one attaches a 2-handle along the simple closed curve associated to µ, and then fills in the remaining S 2 boundary with a 3-ball. The resulting manifold, which we denote by Y (µ) or Y (n/n ′ ), has homology , on which one can now perform Dehn surgery. Whereas our conventionial choice of basis for Dehn filling slopes involves a canonical (up to sign) longitude l, with m (satisfying m · l = 1) only determined up to addition of copies of l, the conventional basis for Dehn surgery involves a canonical meridian, namely µ l , for the knot K µl ⊂ Y (µ l ), with the longitude λ l ∈ H 1 (∂Y ) (satisfying µ l · λ l = 1) only determined up to the addition of copies of µ l .
Thus, for an arbitrary slope, say we could describe the Dehn filling Y (µ) as the n/n ′ -filling of Y (with respect to the basis (m, l)), or as the α/β-surgery along the knot K µl (with respect to the basis (µ l , λ l )). Note that each of these conventional descriptions involves either a denominator or a numerator which is non-canonical. To dodge this problem, we can instead divide the canonical numerator of n/n ′ by the canonical denominator of α/β to obtain n/β, with is not surjective, having determinant p. Still, since this map is injective, it is sufficient for cataloguing slopes. In fact, the reciprocal β/n is more convenient for this purpose. Given an initial filling Y (µ l ) on which we wish to perform surgery, we call (µ l · µ)/(µ · l) = β/n the surgery µ l -label (or just surgery label) of µ. Since (4) n n ′ = p q + β/n , the surgery µ l -label of µ quantifies the deviation of the Dehn filling slope of µ from that of µ l , with a surgery label of β/n = 0 labeling the original slope µ l . We also need conventions for H 1 (Y ), relative to the map ι : In other words, any generatorm for H 1 (Y )/T will satisfy ι(m) ∈ ±gm+ T . We shall always choosem so that ι(m) ∈ +gm + T .

Conventions for Turaev torsion and
where τ (Y ) is the Turaev torsion of Y , which we always normalize so that with t := [m] for any generatorm of When Y is Floer simple, we can also define the torsion complement, with the Floer simplicity of Y guaranteeing that We shall often want to restrict our attention to the non-torsion elements of D τ (Y ), Although we shall not need the following fact until the proof of Theorem 6.2 in Section 6, we lastly remark that the complement of D τ (Y ) is a semigroup.
In the case that Y Floer simple is the complement of the link of a complex planar singularity, Γ(Y ) coincides with the semigroup associated to the Newton-Puiseux expansion.
In terms of our truncation notation,

4.4.
Restating Theorem 1.6 as Theorem 4.2. We are now equipped to re-express Theorem 1.6 in a more practical form, describing the L-space slope interval in terms of any given slope from the interior of that interval, using the "surgery label" description of slopes.
Since the interval of L-space surgery labels always excludes ∞-its being the surgery label of the canonical longitude-we can always describe the interval of L-space surgery labels in terms of its minimum and maximum in Q.
is an L-space if and only if µ / ∈ l , i.e., when µ has finite surgery label.
Remark. It is often more natural to state the above result exclusively in terms of D τ (Y ).
That is, if Y is Floer simple and µ l is an interior L-space slope, then the L-space interval L(Y ) is the smallest interval containing µ l with endpoints in ι −1 (D τ (Y )). This interval is open if its endpoints are equal, and closed otherwise. Of course, one could express the above criterion in any other basis. To characterize Lspace slopes in terms of conventional surgery coefficients, for surgery along the knot core K µl ⊂ Y (µ l ) \ Y associated to a given interior L-space slope µ l = pm + ql, one must first choose a longitude, say λ l := q * m + p * l, with µ l · λ l = 1 implying pp * − qq * = 1. Next, for each δ ∈ D τ >0 (Y ), we express the liftsδ + ,δ − ∈ ι −1 (δ) flanking µ l as ( 18) and Theorem 4.2 takes the following form.
, then for any longitude λ l = q * m+p * l (with µ l ·λ l = 1), the α/β surgery along K µl ⊂ Y (µ l )or equivalently, the Dehn filling Y (µ) with µ := αµ l + βλ l -is an L-space if and only if where ι(a δ In such case, the left hand inequality obtains when β/n < 0, the right hand when β/n > 0, and we regard both inequalities as vacuously true when β/n = 0, where n := µ · l = αp + βq * . If is an L-space if and only if n = 0. One could also characterize L-space slopes in terms of the Dehn filling basis, m, l. If we take µ l = pm+ql to be an interior L-space slope with p > 0, then for any δ = δι(m)+γι(l) ∈ D τ >0 (Y ), it follows from the two identities in (13) that from which it follows that the liftsδ + ,δ − ∈ ι −1 (δ) adjacent to µ l take the form As expected, these are the lifts of δ with Dehn filling slope closest to p/q (regardless of whether p > 0), and Theorem 4.2 takes the following form.
, I δ is the closed interval in Q ∪ {∞} which exludes 0 and has endpoints is an L-space if and only if n = 0.
Example. Suppose K ⊂ S 3 is a Floer simple knot of positive genus g(K), with Alexander polynomial ∆(K). Then Y := S 3 \ ν(K) is Floer simple, and since K ⊂ S 3 Floer simple implies deg ∆(K) = 2g(K), the hypothesis g(K) > 0 implies D τ >0 (Y ) = ∅. Since H 1 (Y ) is torsion free, the endpoints of I δ reduce to δ/ q p and δ/ q p for each δ = δι(m) ∈ D τ >0 (Y ). We already know that the infinity filling Y (1m + 0l) = S 3 is an L-space. Thus (if necessary replacing K with its mirror and using − n n ′ for n n ′ in (25)), we know that Y (pm + 1l) is an L-space for any p > 0 sufficiently large. Taking p > max δ∈D τ >0 (Y ) δ then makes the endpoints of each I δ become δ/ 1 p = δ and δ/ 1 p = +∞, and we recover the well known result that for n ′ = 0, Y (nm + n ′ l) is an L-space if and only if 4.5. Set-up for proof of Theorem 4.2. We begin by making some simplifying assumptions, without loss of generality.
Proposition 4.5. Suppose that Y is Floer simple, that µ l = pm + ql is an L-space slope, and that we wish to determine if µ = nm + n ′ l is an L-space slope for Y . For purposes of proving Theorem 4.2, we may assume, without loss of generality, that p, β > 0, n = 0, pg > deg Proof. Theorem 4.2 already correctly characterizes the cases of β = 0, corresponding to the Dehn filling Y (µ l ), which we already know to be an L-space, and n = 0, for which the filling Y (l) is not a rational homology sphere, hence not an L-space. Likewise, we know that any L-space slope µ l = pm + ql must have p = 0. Since we are free to replace µ l with −µ l or µ with −µ, we may take p, β > 0 without loss. Lastly, by Lemma 3.5, we can approximate µ l with a primitive L-space slope µ ′ l = p ′ m + q ′ l (with q ′ = 0) such that p ′ g > deg [m] τ c (Y ) and gcd(p ′ g, β ′ ) = 1, where β ′ := µ ′ l · µ. We henceforth consider the assumptions of Proposition 4.5 to hold. Given such initial data, we have a primary tool from Heegaard Floer homology to determine whether µ is an L-space slope for the Floer simple manifold Y : namely, Proposition 3.6. To exploit this proposition, we must exhibit Y (µ) as zero surgery on an L-space, given the L-space slope µ l for Y . Fortunately, a standard such construction exists, whereby we first express Y (µ) as some α/β-surgery on Y (µ l ), and then reexpress this as a zero surgery on a connected summed knot inside Y (µ l )#L(β, α * ), for some α * ≡ −α −1 (mod β).

4.7.
Applying the "coloring condition" of Proposition 3.6. Since this section uses the Euler characteristic of knot Floer homology, which we express in terms of the Turaev torsion, regarded as an element of the Laurent series group ring of homology, we briefly introduce generatorsm,m 1 , andm 2 for H 1 (Y # )/T , H 1 (Y )/T 1 , and H 1 (S 3 \ K u ), respectively, with signs chosen so that For notational brevity, we also set In order to use Proposition 3.6, we need the support of the Euler characteristic of the knot Floer homology of K # ⊂ Y # (λ # ). Since HF K tensors on connected sums, its Euler characteristic χ hfk turns tensor product into multiplication, and the support function S[·] on (Laurent series) group rings converts this multiplication of polynomials into addition of sets, yielding

Proposition 2.1 tells us that
is the Turaev torsion, τ c (Y ) is the torsion complement as defined in (8), and we used our simplifying assumption that deg t1 τ c (Y ) < pg. Similarly, we have

Thus, if we set
A then in the language of Proposition 3.6, we have Using the fact that ι(µ # ) = βf 2 ι 2 (m 2 ), one can easily verify that Suppose the above set is nonempty, hence contains some element bι(λ # ) such that >0 (Y ) and evaulating f 1 , f 2 , and ι(λ # ) as expressed in (32) One can use the identity pp * − qq * = 1 to solve the above two equations simultaneously for b, obtaining b ≡ pγ − qδ (mod g). Moreover, taking the first equation modulo p implies b ≡ −qδ (mod p). Thus any solution in b to (46) must satisfy b ≡ pγ − qδ (mod pg).

Seifert Fibered L-spaces
To illustrate the usage of our new L-space interval tool D τ , in this section we exploit Theorem 4.2 to offer a simple alternative proof of a known result: namely, the classification of Seifert fibered spaces over S 2 which are L-spaces. We restrict to the S 2 case because it is the most interesting one, as no higher genus Seifert fibered spaces are L-spaces, and all oriented Seifert fibered spaces over RP 2 are L-spaces [7]. 5.1. Seifert fibered L-spaces, a history. Up until now, the classification of Seifert fibered L-spaces has relied, at least in one direction, on the classification of oriented Seifert fibered spaces M over S 2 admitting transverse foliations, a problem which dates back at least to 1981, when Eisenbud, Hirsch, and Neumann [11] re-expressed this foliations problem in terms of a criterion on representations of π 1 (M ) in Homeo + S 1 , the universal cover of the group of orientation-preserving homeomorphisms of S 1 .
A few years later, Jankins and Neumann [24] reformulated the criterion of [11] in terms of Poincaré's "rotation number" invariant on Homeo + S 1 , a development which, along with the correct conjecture that this criterion is met in Homeo + S 1 if and only if it is met in a smooth Lie subgroup thereof, allowed them to write down an explicit characterization of Seifert fibered manifolds over S 2 admitting transverse foliations. With the exception of one special case, they also showed that this list was complete. It took more than a decade before Naimi [33] resolved this outstanding case using dynamical methods, and more than a decade after that before Calegari and Walker [9] generalized Naimi's methods to provide a proof of the Jankins-Neumann classification that did not appeal to smooth Lie subgroups.
In the late 1990's, Eliashberg and Thurston [12] proved that one can associate a weakly symplectically fillable contact structure to any C 2 cooriented taut foliation on a closed three-manifold-a result which Kazez and Roberts [28], and independently Bowden [5], have recently extended to C 0 foliations. Since Ozsváth and Szabó have [38] shown that this contact structure gives rise to a nontrivial class in Heegaard Floer homology, this proves that L-spaces do not admit co-oriented taut foliations.
In the converse direction, Lisca and Matić [31] proved that a Seifert fibered manifold M over S 2 admits contact structures in each orientation which are transverse to the fibration if and only if M belongs to the explicit set characterized by Jankins and Neumann. Lisca and Stipsicz then showed [32] that if there is an orientation on a Seifert fibered manifold M over S 2 for which no positive contact structure is transverse to the fibration, then M is an L-space.
Since our own answer matches that of Jankins and Neumann, one could take the non-L-space/transverse-foliation equivalence for Seifert fibered manifolds over S 2 as a corollary of Theorem 5.1 below. As for our L-space classification itself, however, the proof no longer requires foliations, dynamical methods, or even (after the proof of Theorem 4.2) contact or symplectic geometry. It only uses ordinary homology and one computation of Turaev torsion from a homology presentation.

Conventions and bases.
To construct a Seifert-fibered space with n exceptional fibers over S 2 , we start with the trivial circle fibration S 1 × S 2 , and remove n + 1 solid tori, . . , n}, yielding a trivial circle fibration over the n + 1-punctured sphere, where ∂ iŶ denotes the i th toroidal boundary component, ∂ iŶ := −∂(S 1 × D 2 i ). Next, we choose presentations for H 1 (Ŷ ) and H 1 (∂ iŶ ) in terms of the regular fiber class f ∈ H 1 (Ŷ ) and classes horizontal to this fiber. For each i ∈ {0, . . . , n}, we take (f i , −h i ) as a reverse-oriented basis for H 1 (∂ iŶ ). Here,h i ∈ H 1 (∂ iŶ ) denotes the meridian of the excised solid torus S 1 × D 2 i , and if we writeι i : , we note that there must be a relation among the h i , since the n + 1-punctured sphere is the same as the n-punctured disk, with first betti number n. In fact, since any one of the h i can be regarded as the class of minus the boundary of this disk, with the remaining h i summing to a class equal to the boundary of the disk, we have n i=0 h i = 0, so that H 1 (Ŷ ) has presentation To specify a Seifert fibered space, one simply lists the Dehn filling slopes, in terms of the basis (f i , −h i ) for each H 1 (∂ iŶ ), of the n + 1 toroidal boundary components ofŶ , conventionally filling ∂ 0 Y with an integer slope and the remaining ∂ i Y with noninteger slopes. That is, for any e 0 , r 1 , . . . , r n ∈ Z and s 1 , . . . , s n ∈ Z =0 with each ri si / ∈ Z, the Seifert Note that for any (z 0 , . . . , z n ) ∈ Z n+1 satisfying n i=0 z i = 0, the change of basis h i → h i + z i f , i ∈ {0, . . . , n}, yields the reparameterization where s denotes the least common positive multiple of s 1 , . . . , s n .
Remark. If we take each s i > 0, then inequality (56) is equivalent to the condition that (57) min The middle expression, e 0 + n i=1 ri si , is the orbifold Euler characteristic. If e 0 + n i=1 ri si = 0, then (57) fails to hold when n ≤ 2, in which case all three expressions are equal.
Theorem 5.1 makes it easy to deduce the L-space filling slope interval for any regular-fiber complement in a Seifert fibered space. That is, for any j ∈ {1, . . . , n}, the above theorem implies that M (e 0 ; r1 s1 , . . . , rn sn ) is an L-space if and only if for all x ∈ {1, . . . , s − 1}. Since the above expressions are integers, (58) holds if and only if Dividing both sides by x then gives the following result.
where s is the least common multiple of those s i with i = j.
Thus ι 0 (l) ∈ H 1 (Y ) is torsion, and so l is also a canonical longitude. Moreover, since gι 0 (l) = n i=1 s siι i (µ i ) = 0 is a primitive linear combination of the relations in the presentation of H 1 (Y ) in (62), we have g = | ι 0 (l) |. Choosing any m ∈ H 1 (∂Y ) satisfying m · l = 1, and writing m = −qf 0 − p * h 0 , allows one to solve forf 0 andh 0 in terms of m and l. Now, since all ri si > 0 by assumption, we know from Ozsváth and Szabó in [37] that Y (−h 0 ) = M (0; r1 s1 , . . . , rn sn ) is an L-space, so we may take µ l := −h 0 as our given L-space filling slope, and choose λ l =f 0 for its longitude, with µ l · λ l = −h 0 ·f 0 = 1. We then have with p and q * as in (64), and with q and p * solving the diophantine equation pp * − qq * = 1.
In such case, we have y i = [r i x] si and z i = rix si for each i ∈ {1, . . . , n}.
We can therefore parameterize This makes a j− x µ l + b j− x λ l ∈ ι −1 0 (δ j x ) one of the two lifts of δ j x closest to µ l in P(H 1 (∂Y )), and the closest lift of δ j x on the other side of µ l is a j+ To use Corollary 4.3 on M (e 0 ; r1 s1 , . . . , rn sn ) = Y (µ 0 ), we shall also want the (µ l , λ l )-surgery coefficients for µ 0 , and the value of µ 0 · l. Since µ 0 = e 0f0 −h 0 and l = pf 0 + q * h 0 , with µ l = −h 0 , λ l =f 0 , p = s g n i=1 ri si , and q * = s g , we have µ 0 = αµ l + βλ l , α := 1, β := e 0 , (83) Since Y (µ 0 ) is never an L-space when µ 0 · l = 0, and since the case of e 0 + n i=1 rix si = 0 is treated separately in the theorem statement, we henceforth restrict to the case of µ 0 · l = 0.
Lastly, suppose that D τ >0 (Y ) = ∅. Since we have excluded the case of µ 0 · l = 0, this implies that Y (µ 0 ) = M (e 0 ; r1 s1 , . . . , rn sn ) is an L-space, so we must show that (87) holds. To see this, first note that the negation of (87) is equivalent to the inequality (88) min . . , n − 1} and x ∈ {1, . . . , s − 1}, we have  (87) is invariant under any map ri si → dri dsi with d ∈ Z =0 , or under any reparameterization of the type in (54), we can remove our initial restrictions that 0 < r i < s i and gcd(r i , s i ) = 1, completing the proof of the theorem.

Gluings along torus boundaries
The introduction to Section 5 discusses how, for Seifert fibered spaces over S 2 (although the same is true for all Seifert fibered spaces [7,14]), the property of admitting a cooriented taut foliation is equivalent to the property of not being an L-space.
6.1. Equivalent properties for Seifert fibered spaces. In fact, this pair of equivalent properties belongs to a larger list. (3) M is not an L-space.
The above result motivated a conjecture of Boyer, Gordon, and Watson [7] that properties (2) and (3) above are equivalent for all closed, prime, oriented three-manifolds.

6.2.
Gluing results. To further explore the relationship of the above properties, Boyer and Clay [6] studied how each of these properties glue together when one splices together Seifert fibered spaces along the toroidal boundaries of fiber complements to form a graph manifold. In the process, Boyer and Clay observed that properties (1) and (2) obey a similar criterion determining when they admit compatible gluings. The property (3) of being a non-L-space proved less tractable for this exercise, but Boyer and Clay conjectured that property (3) should follow a similar gluing pattern to that of (1) and (2).
We are now able to confirm their conjecture in the case in which two Floer simple manifolds glued along their torus boundaries have the interiors of their L-space intervals overlap via the gluing map. In fact, there is no requirement that these Floer simple manifolds be graph manifolds. Theorem 6.2. Suppose that Y 1 and Y 2 are Floer simple manifolds glued together along their boundary tori. Such gluing is specified by a linear map ϕ : H 1 (∂Y 1 ) → H 1 (∂Y 2 ) with det ϕ = −1, descending to a map ϕ P : P(H 1 (∂Y 1 )) → P(H 1 (∂Y 2 )) on Dehn filling slopes. Let I i ⊂ P(H 1 (∂Y i )) denote the interval (with interiorİ i ) of L-space filling slopes for Y i , for each i ∈ {1, 2}, and suppose that ϕ P (İ 1 ) ∩İ 2 is nonempty.
6.5. Computation of D τ (Y ). For the remainder of Section 6, we regard the entire preceding construction, along with the hypotheses of Theorem 6.2, as fixed initial data. We are now ready to compute D τ (Y ), which we shall call D τ ≥0 (Y ) to emphasize that in this case we are not excluding torsion elements. Proposition 6.4. Suppose that µ 1 is "judiciously chosen" from P −1 (İ 1 ∩ϕ −1 P (İ 2 )) nonempty, and that Y is constructed as above. If we set t ∂ : , we need the Turaev torsion τ (Y ) and torsion complement τ c (Y ). In order to write these down, we first choose generatorsm for Recall that the above condition only constrains the signs ofm andm i . We shall write for the inclusions ofm andm i into their respective group rings. Invoking the standard gluing rules for Turaev torsion yields where eachf i denotes the lift of f i to the Laurent series group ring . (One could also obtain this result by using Proposition 2.1 and the fact that Heegaard Floer homology tensors on connected sums. ) For , and let P and P i denote the Laurent series P := P T /(1 − t) and P i := P Ti /(1 − t i ), the latter with polynomial truncations The torsion complements τ c (Y ) : On the other hand, since each (1 − [ι i (µ i )])τ (Y i ) has no negative coefficients, it follows from (103) that τ (Y ) has support Using the facts that 0 ∈ S[τ (Y i )] for each i ∈ {1, 2} (as per the convention stated in (7) in Section 4.2) and that where property (106) has made any remaining subsets of . It is straightforward to show that the above A i are equal to those enumerated in the statement of the proposition. 6.6. Computation of L-space interval for Y . Having determined D τ (Y ), we can apply Theorem 4.2 to compute the L-space interval for Y . Proposition 6.5. Suppose that µ 1 is "judiciously chosen" from P −1 (İ 1 ∩ϕ −1 P (İ 2 )) nonempty, and that Y is constructed as above. For each i ∈ {1, 2}, setq i := [q * i ] pi and let B i denote the set is an L-space if and only if condition (l.i) holds for each b 1 ∈ B 1 , (l.ii) holds for each b 2 ∈ B 2 , and (l.iii) where p := p 1 p 2 and g := g 1 g 2 /g 0 , with g 0 = gcd(g 1 , g 2 ).
for any liftδ ∈ ι −1 (δ), there always exists a unique a δ ∈ Z for which δ = ι(a δ µ l + b δ λ l ). Such a δ ∈ Z satisfies δ = a δ p + b δ q * . Taking this as a definition for a δ ∈ Z, we note that, since b δ − p < 0 and q * > 0, the left-hand inequality in (112) is vacuous, whereas the right-hand inequality is equivalent to the condition a δ ≤ 0.

6.7.
Determining when gluing hypothesis is met. We next turn our attention to the L-space filling slope intervals I i ⊂ P(H 1 (∂Y i )), to determine when they combine according to the hypotheses of the theorem.
6.8. Comparison of L-space classification with gluing hypothesis. Now that we have both classified when Y 1 ∪ ϕ Y 2 is an L-space, and classified when it satisfies the gluing hypothesis in terms of the union of the L-space intervals of Y 1 and Y 2 , it remains to show that these two classifications are equivalent. Proposition 6.7. Suppose that µ 1 is "judiciously chosen" from P −1 (İ 1 ∩ϕ −1 P (İ 2 )) nonempty, and that Y is constructed as above. For each i ∈ {1, 2}, setq i := [q * i ] pi and let B i denote the set Then condition (i.i) (respectively (i.ii)) from Proposition 6.6 holds if and only if condition (l.i) (respectively (l.ii)) from Proposition 6.5 holds for all b 1 ∈ B 1 (respectively b 2 ∈ B 2 ).
We now proceed with an inductive argument. Suppose that (l.iii) holds for all (b 1 , b 2 ) ∈ B 1 × B 2 satisfying b 1 ≡ b 2 (mod g 0 ), and that (i.i) and (i.ii) hold, but that there exist b i ∈ B i and b I ∈ B I , with {i, I} = {1, 2} and b i ≤ b I , for which (i.iii) fails, i.e., for which Equation (139) from our Claim then tells us that This means that b i / ∈ B I , since otherwise, setting b := b i ∈ B i ∩ B I = B 1 ∩ B 2 would make (148) contradict condition (l.iii). Thus, δ bi I / ∈ D τ ≥0 (Y I ) and b i < b I . so the Milnor torsion is More generally, the same argument shows that Motivated by this, we make the following If Y is such a manifold, Corollary 2.3 implies that Y ≤ g Y − 2. On the other hand, an embedded surface which generates H 2 (Y, ∂Y ) has at least g Y boundary components, so a norm-minimizing surface must have genus 0.
The Milnor torsion of a generalized solid torus is determined by g Y and k Y .
Proof. The usual product formula for the torsion implies that where c ∈ Z and p(t) ∈ Z[t ±1 ]. |H 1 (Y (l))| = k Y . Combining the two formulas, we see that Combining the lemma with the requirement that deg ∆(Y ) < g Y gives In contrast, τ (Y ) is not determined by the fact that Y is a generalized solid torus, as can be seen by considering the Seifert-fibred spaces M (∅; a/g, −a/g). Proof. Let g = g Y . Recall that Y is a Floer homology solid torus if CF D(Y, m, l) ≃ CF D(Y, m + l, l), where l is the canonical longitude and m · l = 1. By composing with an appropriate change of basis bimodule, we see that this is equivalent to saying that for some µ, λ with µ · λ = 1, we have CF D(Y, µ, λ) ≃ CF D(Y, τ l (µ), τ l (λ)), where τ l is the Dehn twist along l.
Suppose that Y is a generalized solid torus. By Proposition 3.9, we can explicitly compute CF D(Y, µ, λ) for an appropriate choice of µ and λ. In fact, CF D(µ, λ) is determined by the polynomials χ( HF K(K µ )) and χ( HF K(K λ )), which are in turn determined by ∆(Y ), ι(µ), and ι(λ). Since Y = g − 2, the criteria of Proposition 3.9 will be satisfied if we take µ = m and λ = l − N m, where N ≫ 0.
We define an isomorphism f : CF D(Y, µ, λ) → CF D(Y, µ ′ , λ ′ ). The map f : HF K(K µ ) → HF K(K µ ′ ) is given as follows. If x ∈ S µ , then f takes the unique nonzero element of HF K(K µ ) supported at x to the unique nonzero element of HF K(K µ ′ ) supported at x + ⌊φ(x)/g⌋l. Using the description of the sets S µ and S ′ µ given above, together with the fact that φ(µ) = g, it is easy to see that f is a bijection. Similarly, if x ∈ S λ , we define f to take the unique nonzero element supported at x to the unique nonzero element of HF K(K λ ′ ) supported at x + ⌊φ(x)/g⌋l.
It remains to check that f carries the arrows in the diagram for C = CF D(Y, µ, λ) to the arrows in the diagram for C ′ = CF D(Y, µ ′ , λ ′ ). Suppose x and y are the initial and terminal ends of an arrow of type D 23 in C, so that y − x = µ. Then φ(y) − φ(x) = g, so f (y) − f (x) = µ + l = µ ′ , so f (y) and f (x) are the endpoints of an arrow of type D 23 in C ′ . A very similar argument shows that arrows of types D 1 and D 3 are preserved as well.
We can prove a partial converse to Proposition 7.1. Recall that Y is said to be semiprimitive if T ⊂ im ι. Equivalently, Y is semi-primitive if k Y = 1.
Proposition 7.6. Suppose that Y is semi-primitive and Floer simple. If D τ >0 (Y ) = ∅, then Y is a generalized solid torus.
Proof. Let g = g Y . Since Y is semiprimitive, we have H 1 (Y ) = Z ⊕ (Z/g) and also im ι = gZ ⊕ Z/g ⊂ H 1 (Y ). Let t, σ be generators of the Z and Z/g summands respectively, so that τ (Y ) = ∞ i=0 q i (σ)t i , where q i (σ) is a sum of powers of σ. Suppose that for some value of i, q i (1) < g and q i−g (1) > 0. Then we can find x ∈ S[τ (Y )] with φ(x) = i and y ∈ S[τ (Y )] with φ(y) = i − g. It follows that x − y ∈ im ι, which contradicts D τ >0 (Y ) = 0. We conclude that for a fixed value of k there is at most one value of n for which q k+ng (1) = 0, g.
Lemma 7.7. There is a constant c so that i≡k (g) a i ≡ k + c (g).
Note that all but finitely many of the a i are equal to either 0 or g, so the sum is well defined.
Proof. We say that f (t) ∈ Z[t] has property (*) if the statement of the corollary holds for a i given by f (t)/(1 − t) = ∞ i=0 a i t i . It is easy to see that f (t) = 1 + t + . . . + t g−1 has property (*), and that if f (t) has property (*), then so do f (t) + t i − t g+i and t c f (t). Lemma 7.3 implies that ∆(Y ) can be obtained from 1 + t + . . . + t g−1 by a sequence of operations of the first type plus a single operation of the second type, so ∆(Y ) has property (*).
The lemma implies that after an appropriate shift in the indexing of the a i 's (so that τ (Y ) is no longer constrained to to have t 0 as its lowest order term) the subsequence (a k+ng ) has the form . . . , 0, 0, 0, k, g, g, g . . ., where 0 ≤ k ≤ g. In other words, each subsequence is determined up to a global shift, and it remains to see how these shifts fit together.
To see this, let us say that Q(t) ∈ Z[t −1 , t]] is obtained from P (t) by an elementary shift if Q(t) − P (t) = at i + (g − a)t i+g for some a, i ∈ Z. We have shown above that τ (Y ) is obtained from τ 0 by a sequence of elementary shifts. Next, we consider the effect of an elementary shift on the Alexander polynomial. If Q(t) ∈ Z[t −1 , t]], let F (Q(t)) = p((1−t)Q(t)), where p : Z[t] → Z[t]/(t g −1) is the projection, so that F (τ 0 ) = 1+. . .+t g−1 . An easy calculation shows that if Q(t)−P (t) = at i +(g−a)t i+g , then F (Q(t))−F (P (t)) = gt i −gt i+1 . It follows that if Q(t) is obtained from τ 0 by a sequence of elementary shifts and F (Q(t)) = F (τ 0 ), then Q(t) is obtained from τ 0 by a global shift; that is, each residue class is shifted by the same number of elementary shifts. To sum up, we have proved that τ (Y ) ∼ τ 0 , so Y is a generalized solid torus.
As we observed above, if Y is a generalized solid torus, H 2 (Y, ∂Y ) is generated by a surface of genus 0. It follows that Y (l) = Z#(S 1 × S 2 ), where Z is a rational homology sphere. Conversely, we have Proposition 7.8. Suppose that K ⊂ Z#(S 1 × S 2 ) has an L-space surgery. Then the complement of K is a generalized solid torus.
Proof. We use the exact triangle with twisted coefficients, as formulated by Ai and Peters in [1]. We briefly recall their statement. Given a class η ∈ H 1 (Y ) and µ ∈ Sl(Y ), we can form ω µ = P D(j * (η)) ∈ H 2 (Y (µ)), where j : Y → Y (µ) is the inclusion. The twisted Floer homology HF (Y (µ); Λ ωµ ) is a module over the universal Novikov ring Λ = a r t r | r ∈ R, a r ∈ Z, #{r < C | a r = 0} < ∞ for all C ∈ R .
Let Y be the complement of K, so Y (l) = Z#(S 1 × S 2 ). Choose η ∈ H 1 (Y ) with φ(η) = 1, so that ω l generates H 2 (Y (l)) = Z. By [1] Proposition 2.2, HF (Y (l); Λ ω λ ) = 0. Now suppose there is some m with m · l = 1 and m ∈ L(Y ). In this case H 2 (Y (m); R) ≃ H 2 (Y (m + l); R) = 0. The exact triangle shows that HF (Y (m)) ⊗ Λ ≃ HF (Y (m + l)) ⊗ Λ, which implies that HF (Y (m)) ≃ HF (Y (m + l)). Since H 1 (Y (m)) ≃ H 1 (Y (m + l) ), it follows that m + l ∈ L(Y ). Repeating, we find that m + nl ∈ L(Y ) for all n > 0, and thus that l is a limit point of L(Y ). It follows that Y is Floer simple and D τ >0 (Y ) = ∅. For the general case, suppose that µ ∈ L(Y ). Then Y (l) is obtained by integer surgery on K µ #K −q/p ⊂ Y (µ)#L(q, −p) for an appropriate choice of p and q. Let Y ′ be the complement of this knot. The argument above shows that every non-longitudinal filling of Y ′ is an L-space. An infinite family of these fillings are also obtained by Dehn filling on Y , so Y is Floer simple.
Boyer and Clay define α to be NLS detected if it is NLS detected by some N g , where N g = M (1/g, 1 − /g) is the original family of Floer homology solid tori discussed above. The proposition shows that α is NLS detected by one N g if and only if it is NLS detected by all N g if and only if α is not the interior of L(Y ). This proves Corollary 1.12. 7.3. Examples. We conclude by constructing some examples of generalized solid tori. Some of these were previously known to Hanselman and Watson [18] and Vafaee [46]. We start with the following observation.
Corollary 7.12. If Y is an irreducible, semi-primitive generalized solid torus, then Y is the complement of a closed g Y -strand braid in S 1 × S 2 .
Proof. The hypotheses imply that ∆(Y ) ∼ (1 − t g )/(1 − t) and that H 2 (Y, ∂Y ) is generated by a g Y -times punctured sphere. By Corollary 2.3, it follows that Y fibres over S 1 with fibre of genus 0. By Proposition 7.8, any knot in S 1 × S 2 with a lens space surgery is a generalized solid torus. Cebanu [10] showed that a knot of this form is a closed braid in S 2 . Examples of such knots were studied by Buck, Baker, and Leucona in [2]. Many (but not all) of them are derived from knots in the solid torus which have solid torus surgeries. These knots were completely classified by Gabai [15] and Berge [3].
To find other examples, we look for braids in S 1 × S 2 which have L-space surgeries. One criterion for finding such examples is given here. Suppose σ is an ordinary g strand braid in D 2 × I. We can close σ to get a closed braid in S 1 × D 2 . Dehn filling S 1 × D 2 along S 1 × p gives the ordinary braid closure σ ⊂ S 3 . We can also fill S 1 × D 2 along ∂D 2 to get a closed braid in S 1 × S 2 , which we denote by σ. Let ∆ ∈ Br g be the full twist on g-strands.
Proposition 7.13. Suppose that σ is a braid with the property that K n = ∆ n σ is an Lspace knot in S 3 for all n ≥ 0. Then the complement of σ is a semi-primitive generalized solid torus.
Proof. Let L ⊂ S 3 be the link which is the union of K = σ and the braid axis B. The braid σ is the image of K in the S 1 × S 2 obtained by doing 0-surgery on B.
Let L(a, c) be the manifold obtained by doing a surgery on K and c surgery on A, where a ∈ Z and c ∈ Q. Then L(a, −1/n) is the result of a + ng 2 surgery on K n . Using Seifert's algorithm, it is easy to see that there is a constant C(σ) with the property that g(K n ) ≤ C(σ) + ng(g − 1)/2. Thus if a > 2C(σ), then a + ng 2 ≥ 2g(K n ) − 1 for all n ≥ 0. By hypothesis, K n is a positive L-space knot, so L(a, −1/n) is an L-space for all n > 0. Now let Y be the manifold obtained by doing a surgery on K, and let Y = Y − ν(B). There is a slope α 0 ∈ Sl(Y ) so that Y (α 0 ) = L(a, 0), and a sequence of slopes α −1/n ∈ Sl(Y ) which converge to α 0 such that Y (α −1/n ) = L(a, −1/n). It follows that Y is Floer simple and that α 0 is in the closure of L(Y ). Since α 0 is not the homological longitude of Y , α 0 ∈ L(Y ), so L(a, 0) is an L-space. By Proposition 7.8, Y is a generalized solid torus.
We call a closed braid in the solid torus which satisfies the criterion a L-space braid. Examples include: • Knots in the solid torus with solid torus surgeries (aka Berge-Gabai knots) • The twisted torus knots T (p, kp ± 1; 2, 1) studied by Vafaee [47] • Cables of L-space braids [21] • Satellites where the pattern knot is a Berge-Gabai knot and the companion is an L-space braid [22] We conclude with two remarks. First, we conjecture that every positive one-bridge braid (not just the Berge-Gabai knots) is an L-space braid. Since the knot obtained by applying a full twist to a one-bridge braid is again a one-bridge braid, this is equivalent to showing that the closure of any positive one-bridge braid is an L-space knot in S 3 . Second, in light of the last two items, it would be interesting to know if a satellite where both the pattern and the companion are L-space braids is also an L-space braid.