Floer Simple Manifolds and L-Space Intervals
aa r X i v : . [ m a t h . G T ] N ov FLOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS
JACOB RASMUSSEN AND SARAH DEAN RASMUSSEN
Abstract.
An oriented three-manifold with torus boundary admits either no L-spaceDehn 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 , and prove a special case of a conjecture of Boyer and Clay [6] about L-spaces formedby gluing three-manifolds along a torus. Introduction
An oriented rational homology 3-sphere Y is called an L-space if the Heegaard Floerhomology d HF ( Y ) satisfies d HF ( Y, s ) ≃ Z for each Spin c structure s on Y . Recent interest inthe 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 andonly if π ( Y ) is non left-orderable. Subsequently, Boyer and Clay [6] studied a relativeversion 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 singletorus boundary component. If Y is such a manifold, we let Sl ( Y ) = { α ∈ H ( ∂Y ) | α is primitive } / ± ∂Y . Sl ( Y ) is a one-dimensional projective space defined over therational numbers. If we fix a basis h µ, λ i for H ( 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 Dehnfilling Y with slope α , and let K α ⊂ Y ( α ) be the core of the filling solid torus. Definition 1.1. If Y is a compact connected oriented three-manifold with torus boundary, L ( Y ) = { α ∈ Sl ( Y ) | Y ( α ) is an L-space } is the set of L-space filling slopes of Y . For the set L ( Y ) to be nonempty, we must have b ( Y ) = 1, which implies that Y isa rational homology S × D . In this paper, we will restrict our attention to manifoldswith multiple L-space fillings: that is, for which |L ( Y ) | >
1. Such manifolds can be easilycharacterized in terms of their Floer homology. Recall that a knot K in a rational homologysphere Y is Floer simple [20] if the knot Floer homology \ HF K ( K ) ≃ Z | H ( Y ) | . Equivalently, K is Floer simple if Y is an L-space and the spectral sequence from \ HF K ( K ) to d 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 ) . JR was partially supported by EPSRC grant EP/M000648/1.SDR was supported by EPSRC grant EP/M000648/1.
Then we have
Proposition 1.3. |L ( Y ) | > if and only if Y is Floer simple. If K α ⊂ Y ( α ) is Floer simple, then the Floer homology of any surgery on K α canbe determined from \ HF K ( K α ) using the Ozsv´ath-Szab´o mapping cone. The knot Floerhomology, in turn, is determined by the Turaev torsion τ ( Y ) via the relation χ ( \ HF K ( K α )) ∼ (1 − [ α ]) τ ( Y )established in Proposition 2.1. It follows that if Y is Floer simple, then the Floer homologyof 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 ( Y ) = Z ⊕ T , where T is a torsion group, and let φ : H ( Y ) → Z be theprojection. Properly normalized, τ ( Y ) can be written as a sum τ ( Y ) = X h ∈ H ( Y ) φ ( h ) ≥ a h [ h ] , where a h = 1 for all but finitely many h ∈ H ( Y ) with φ ( h ) >
0, and a = 0. For example,if H ( Y ) = Z , then τ ( Y ) = ∆( Y )1 − t ∈ Z [[ t ]] , where the Alexander polynomial ∆( Y ) is normalized to be an element of Z [ t ] and we expandthe denominator as a Laurent series in positive powers of t . Proposition 1.4.
When Y is Floer simple, every coefficient a h of τ ( Y ) is either or . Let S [ τ ( Y )] = { h ∈ H ( Y ) | a h = 0 } denote the support of τ ( Y ), and let ι : H ( ∂Y ) → H ( Y ) be the map induced by inclusion. Definition 1.5. If Y is a Floer simple manifold, we define D τ ( Y ) = { x − y | x / ∈ S [ τ ( Y )] , y ∈ S [ τ ( Y )] , φ ( x ) ≥ φ ( y ) } ∩ im ι ⊂ H ( Y ) , and write D τ > ( Y ) for the subset of D τ ( Y ) consisting of those elements with φ ( h ) > . Let [ l ] ∈ Sl ( Y ) be the homological longitude ( i.e. l is a primitive element of H ( Y ) suchthat ι ( l ) is torsion.) We can now state our first main theorem: Theorem 1.6. If Y is Floer simple, then either D τ > ( Y ) = ∅ and L ( Y ) = Sl ( Y ) \ [ l ] ,or D τ > ( Y ) = ∅ and L ( Y ) is a closed interval whose endpoints are consecutive elements of ι − ( D τ > ( 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 endpointsin ι − ( D τ > ( 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 and Y are rational homology solid tori, and that ϕ : ∂Y → ∂Y is anorientation reversing diffeomorphism. The manifold Y ϕ = Y ∪ ϕ Y is said to obtained bysplicing Y and Y together by ϕ .In [6], Boyer and Clay studied how the presence of structure ( ∗ ) on Dehn fillings of thepieces Y and Y 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 ( Y ϕ ); or3) a nontrivial class in HF red ( Y ϕ ) (as HF red vanishes on, and only on, L-spaces). When Y and Y are graph manifolds, they obtained very strong results in cases 1) and 2), in additionto less complete results in the third case. The analogy with the first two cases suggests thefollowing conjecture, which is implicit in the work of Boyer and Clay and stated explicitlyin certain cases by Hanselman [16]. Conjecture 1.7.
Suppose that Y and Y 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 ◦ ) ∪L ◦ = Sl ( Y ) . In particular, the conjecture says that in order for Y ϕ to be an L-space, both Y and Y must be Floer simple. Our second main result is Theorem 1.8.
Suppose that Y and Y as above are Floer simple and have D τ = ∅ , andthat ϕ ∗ ( L ◦ ) ∪ L ◦ = ∅ . Then Y ϕ is an L-space if and only if ϕ ∗ ( L ◦ ) ∪ L ◦ = Sl ( Y ) . Hanselman and Watson [19] have proved a similar theorem using bordered Floer homol-ogy. The restriction that ϕ ∗ ( L ◦ ) ∩ L ◦ = ∅ represents a limitation of our approach, ratherthan 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 ◦ ) ∩ L ◦ = ∅ , we have noconvenient 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 con-dition on the bordered Floer homology, which they call simple loop type . In a subsequentjoint paper [17], it is shown that the conditions of being Floer simple and being simple looptype are equivalent thus enabling us to remove the hypothesis that ϕ ∗ ( L ◦ ) ∩ L ◦ = ∅ . Theproof of this fact relies on Proposition 3.9 of the current paper, where we explicitly computethe bordered Floer homology \ CF D ( Y, µ, λ ) of a Floer simple manifold Y for an appropriatechoice of µ, λ ∈ H ( ∂Y ) parametrizing ∂Y .We briefly discuss those aspects of Conjecture 1.7 which are not covered by Theorem 1.8and 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 ( Y ) ≃ Z ,or more generally, when Y is semi-primitive ( c.f. Proposition 1.9 below), but in general wehave very little idea how to address this question. (Indeed, this seems like the weakest pointof the conjecture.) The other situation which is not addressed by Theorem 1.8 is the casewhere one or both of Y and Y is not Floer simple. It seems plausible that bordered Floerhomology could be used to prove the conjecture when |L ( Y ) | = 1 and |L ( Y ) | >
1, or when |L ( Y ) | = |L ( Y ) | = 1. In contrast, the case where one or both of the Y i has no L-spacefillings seems considerably more difficult to address with current technology.1.2.
Floer homology solid tori.
The class of Floer simple manifolds with D τ > = ∅ is ofspecial interest. If Y is a rational homology S × D , we say that Y is semi-primitive if thetorsion subgroup of Y is contained in the image of ι , and that Y has genus 0 if H ( 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 > ( Y ) = ∅ . (2) Y is Floer simple and has genus . (3) Y has genus and has an L-space filling. For example, if K ⊂ S × S has a lens space surgery, then the complement of K satisfiesthe 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 arediscussed in section 7.3. JACOB RASMUSSEN AND SARAH DEAN RASMUSSEN
The conditions of Proposition 1.9 are closely related to Watson’s notion of a
Floer homol-ogy solid torus . Suppose that Y is a rational homology S × D with homological longitude l , and that m ∈ H ( ∂Y ) satisfies m · l = 1. Definition 1.10. [18] Y is a Floer homology solid torus if \ CF D ( Y, m, l ) ≃ \ CF D ( Y, m + l, l ) . Proposition 1.11. If Y satisfies the conditions of Proposition 1.9, then it is a Floer ho-mology solid torus. Manifolds with D τ > ( Y ) = ∅ play an important role in the notion of NLS detection intro-duced by Boyer and Clay in [6]. If Y is a rational homology S × D 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 certainsplicings of Y with a family of Floer homology solid tori are not L-spaces. (For the precisedefinition, see section 7.2). By Theorem 1.6, the set of strongly NLS detected slopes is eithera single point, an open interval in Sl ( Y ), or all of Sl ( Y ). By combining Theorem 1.8 withsome direct geometric computation, we can show Corollary 1.12. If Y is a rational homology S × D , the set of NLS detected slopes in Sl ( Y ) is the closure of the set of strongly NLS detected slopes. Seifert fibred spaces.
One of the key motivating examples for the conjecture of [7] isthe class of Seifert-fibred spaces. Indeed, building on work of Ozsv´ath, Szab´o, Mati´c, Naimi,Jankins, Neumann, Eisenbud, and Hirsch [38, 31, 33, 24, 11], Lisca and Stipsicz proved
Theorem 1.13. [32]
A Seifert fibred space over S is an L-space if and only it does notadmit a coorientable taut foliation. In combination with a result of Boyer, Rolfsen, and Wiest [8], this also implies that aSeifert-fibred space over S has non left-orderable π if and only if it is an L-space. Theset of Seifert fibred spaces over S which admit a coorientable taut foliation was explicitlydescribed by Jankins and Neumann [24] and Naimi [33], building on a result of Eisenbud,Hirsch, and Neumann [11].Any Seifert-fibred space over S can be obtained by Dehn filling a Seifert fibred space over D . It follows easily from work of Ozsv´ath and Szab´o [37] that any Seifert fibred space over D 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 agreeswith 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 manifoldsand their relation to the conjecture of Boyer, Gordon, and Watson. First, we recall thestatement of the conjecture.
Conjecture 1.14. [7] If Y is a oriented, closed, prime three-manifold, then Y is an L-spaceif and only if π ( Y ) is non left-orderable. A potentially more tractable subset of this problem, raised by Boyer and Clay [6] is:
Question 1.
Suppose Y is Floer simple. Is π ( Y ( α )) non left-orderable equivalent to α being an element of L ( Y ) ? The characterization of L ( Y ) given in Theorem 1.6 should make it possible to conductmore detailed tests of Conjecture 1.14. Since there is already considerable experimentalevidence in support of the conjecture, we should also consider what circumstances mightexplain a positive answer to Question 1. One possible explanation is that the condition ofbeing Floer simple is correlated with some strong geometrical property, which in turn canbe related to orderings of π . LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 5
Question 2.
Is there a geometric characterization of Floer simple manifolds which can bestated without reference to Floer homology?
More generally, we think that Floer simple manifolds are a natural class of manifoldswhose geometrical properties should be investigated for their own sake. Some evidence insupport of this idea is provided by the frequency of Floer simple manifolds among geo-metrically simple 3-manifolds (as measured by the SnapPea census). Proposition 1.3 maylead readers familiar with the example of L-space knots in S to suspect that the class ofFloer simple manifolds is relatively small, but this is not the case. Of the 59,068 rationalhomology S × D ’s in the SnapPy census of manifolds triangulated by at most 9 idealtetrahedra, nearly 20% have multiple finite fillings, and are thus certifiably Floer simple.Moreover, more than two-thirds of the remaining manifolds have Turaev torsion compatiblewith their being Floer simple. It seems likely that many of these manifolds are Floer simpleas well. (The authors thank Tom Brown for sharing these statistics with them.) For thosewho like other geometries, we note that every Seifert fibred rational homology S × D isFloer simple.It would be interesting to know what happens to the density of Floer simple manifoldsas the complexity increases. Perhaps the most basic question we could ask along these linesis Question 3.
Are there infinitely many irreducible Floer simple manifolds with the sameTuraev torsion?
Organization.
The remainder of the paper is organized as follows. In section 2, wereview some facts about knot Floer homology and the Ozsv´ath-Szab´o mapping cone. Theseare used in section 3 to prove Proposition 1.3 and to give a characterization of when a givensurgery on a Floer simple knot produces an L-space. In this section, we also explain how tocompute the bordered Floer homology of a Floer simple manifold. Theorem 1.6 is provedin In Section 4. In Section 5 we apply Theorem 1.6 to Seifert fibred spaces, thus giving anew proof of Theorem 1.13. The proof of Theorem 1.8 is given in Section 6. Finally, inSection 7, we discuss manifolds with D τ > = ∅ . Acknowledgements:
The authors would like to thank Steve Boyer, Tom Brown, AdamClay, Tom Gillespie, Jonathan Hanselman, Robert Lipshitz, Saul Schleimer, FaramarzVafaee, and Liam Watson for helpful conversations. We also thank the organizers of the 9thWilliam Rowan Hamilton conference in Dublin, which helped to get this project started.2.
Knot Floer homology and the Ozsv´ath-Szab´o mapping cone
In this section, we briefly recall some facts about knot Floer homology [36, 43, 40] whichwill be used in what follows. First, let us fix some notation. Throughout this section, weassume 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 ( ∂Y ) the class of its meridian. Furthermore, welet T ⊂ H ( Y ) be the torsion subgroup, and denote by φ : H ( Y ) → Z the projection from H ( Y ) to H ( Y ) /T ≃ Z , where the isomorphism is chosen so that φ ( µ ) > Knot Floer homology.
The knot Floer homology \ HF K ( K ) is a finitely generatedabelian group with an absolute Z / \ 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 ( Y ) ( aka H ( Y ) torsor); it has a free transitive actionof H ( Y ). The group \ HF K ( K, s ) is trivial for all but finitely many s ∈ Spin c ( Y, ∂Y ). JACOB RASMUSSEN AND SARAH DEAN RASMUSSEN
Given s ∈ Spin c ( Y, ∂Y ), we consider the formal sum χ s ( \ HF K ( K )) := X h ∈ H ( Y ) χ ( \ HF K ( K, s + h ))[ h ] , where χ ( \ HF K ( K, s )) is defined using the absolute Z / χ s ( \ HF K ( K )) asan element of the group ring Z [ H ( Y )]; it is known as the graded Euler characteristic of \ HF K ( K ). Clearly χ s ′ ( \ HF K ( K )) = [ s − s ′ ] χ s ( \ HF K ( K )) . From now on, we will drop s from the notation and view χ ( \ HF K ( K )) as an element of Z [ H ( Y )], well defined up to global multiplication by elements of H ( Y ). We write x ∼ y if x, y ∈ Z [ H ( Y )] satisfy x = [ h ] y for some h ∈ H ( Y ).For knots in S , it is well-known that χ ( \ HF K ( K )) is the Alexander polynomial of K .More generally, we have Proposition 2.1. χ ( \ HF K ( K )) ∼ (1 − [ µ ]) τ ( Y ) , where τ ( Y ) is the Turaev torsion of Y .Proof. \ HF K ( K ) can be identified with the sutured Floer homology SF H ( Y, γ µ ) [25], wherethe suture γ µ consists of two parallel copies of µ . The Euler characteristic of the suturedFloer homology can be described as an appropriately formulated torsion [13]. When ∂Y istoroidal, 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 , but the proof carries through unchanged.) (cid:3) A priori , τ ( Y ) is an element of the field Q ( H ( Y )) obtained by inverting all elements of Z [ H ( Y )] which are not zero divisors. Choose any primitive µ ∈ H ( ∂Y ) with φ ( µ ) = 0;then 1 − [ µ ] will not be a zero divisor in Z [ H ( Y )]. It follows from the proposition that τ ( Y ) ∈ Z [ H ( Y )][(1 − [ µ ]) − ] ⊂ Q ( H ( Y )).Writing (1 − [ µ ]) − = P ∞ i =0 [ µ ] i allows us to embed Z [ H ( Y )][(1 − [ µ ]) − ] in the Novikovring Λ φ [ H ( Y )] = n X h ∈ H ( Y ) a h [ h ] (cid:12)(cid:12)(cid:12) { h | a h = 0 , φ ( h ) < k } < ∞ for all k o . We will view τ ( Y ) as an element of Λ φ [ H ( Y )]. By choosing a splitting H ( Y ) ≃ Z ⊕ T , wecan identify Λ φ [ H ( Y )] with the Laurent series ring Z [ t − , t ]] ⊗ Z [ T ], which we shall latersometimes call the “Laurent series group ring.”As an element of the Novikov ring, τ ( Y ) is well-defined up to multiplication by elementsof H ( Y ). We shall always normalize so that τ ( Y ) has the form τ ( Y ) = P h a h [ h ], where a h = 0 for all h with φ ( h ) <
0, and a = 0.If H ( Y ) = Z , it is well-known that τ ( Y ) ∼ ∆( Y ) / (1 − t ), where ∆( Y ) is the Alexanderpolynomial of Y . More generally, if Φ : Λ φ [ H ( Y )] → Z [ t − , t ]] is the map induced by theprojection φ : H ( Y ) → Z , we define τ ( Y ) = Φ( τ ( Y )) and ∆( Y ) = (1 − t ) τ ( Y ) . Note that in general, ∆( Y ) = ∆( Y ); an interesting example to consider is the connected sum Y = Z S × D ), where b ( Z ) = 0. This manifold has ∆( Y ) = 0, but ∆( Y ) = | H ( Z ) | .If K is a knot in S , it is well known that deg ∆( t ) ≤ g ( K ), and ∆( K ) | t =1 = 1. Thefollowing result is a simultaneous generalization of these two facts. Proposition 2.2 ([45] Lemma II.4.5.1 and Theorem II.4.2.1) . If k Y k is the Thurston normof a generator of H ( Y, ∂Y ) and τ ( Y ) is normalized as above, then a h = 1 for all h ∈ H ( Y ) with φ ( h ) > k Y k . LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 7
More generally, it is known that \ HF K ( K ) determines both the Thurston norm of Y andwhether it is fibred [35, 34, 26]. Since the knot Floer homology of a Floer simple knot isdetermined by its Euler characteristic, we have Corollary 2.3. If Y is boundary incompressible and Floer simple, k Y k = deg ∆( Y ) − . If Y is also irreducible, then Y fibres over S if and only if ∆( Y ) is monic. Differentials.
The knot Floer homology of K can be used to compute the Floer ho-mology of surgeries on K . Before we explain how to do this, we must understand the relationbetween \ HF K ( K ) and d HF ( Y ).We begin by discussing Spin c structures. There are maps i v , i h : Spin c ( Y, ∂Y ) → Spin c ( Y )which respect the action of H ( Y ), in the sense that i v ( s + a ) = i v ( s ) + i ∗ ( a ) and i h ( s + a ) = i h ( s ) + i ∗ ( a ) where i ∗ : H ( Y ) → H ( Y ) is the map induced by inclusion. Moreover, i v ( s ) − i h ( s ) = i ∗ ( λ ), where λ is a longitude of K . We define an equivalence relation onSpin c ( Y, ∂Y ) by declaring s ∼ s if i v ( s ) = i v ( s ). It is easy to see that this is the sameas requiring that i h ( s ) = i h ( s ), and that the equivalence classes are orbits of Spin c ( Y, ∂Y )under the action of µ .Let e s be an equivalence class in Spin c ( Y, ∂Y ). After we choose some auxiliary data (adoubly pointed Heegaard diagram for K ), Heegaard Floer homology constructs for us agraded group \ CF K ( K, e s ) = M s ∈ e s \ CF K ( K, s )together with maps d , d v , d h : \ CF K ( K, e s ) → \ CF K ( K, e s ), which are filtered with respectto the Spin c grading in the following sense: if x ∈ \ CF K ( Y, s ), then d x ∈ \ CF K ( Y, s ), d v x ∈ ⊕ k< \ CF K ( Y, s + kµ ) and d h x ∈ ⊕ k> \ CF K ( Y, s + kµ ) . These differentials satisfy therelations d = ( d + d v ) = ( d + d h ) = 0. Furthermore, we have H ( \ CF K ( K, s ) , d ) = \ HF K ( K, s ) ,H ( \ CF K ( K, e s ) , d + d v ) = \ HF K ( Y , i v ( s )) ,H ( \ CF K ( K, e s ) , d + d h ) = \ HF K ( Y , i h ( s )) . The Spin c grading provides a natural filtration on the latter two complexes, in the sensethat ⊕ k
For each s ∈ Spin c ( Y ) , the bent complex is A K, s = ( \ CF K ( K, e s ) , d s ) ,where for x ∈ \ CF K ( K, s + kµ ) , d s ( x ) = d ( x ) + d v ( x ) k < d ( x ) + d v ( x ) + d h ( x ) k = 0 d ( x ) + d h ( x ) k > . The bent complexes measure the Heegaard Floer homology of large integer surgery on K : H ( A K, s ) ≃ d HF ( Y ( N µ + λ ) , i n ( s )) for sufficiently large N and an appropriately chosenSpin c structure i N ( s ) on the filling. JACOB RASMUSSEN AND SARAH DEAN RASMUSSEN
The existence of the Spin c filtration means there are chain maps π v : A K, s → ( \ CF K ( K, e s ) , d + d v ) π h : A K, s → ( \ CF K ( K, e s ) , d + d h )given by π v ( x ) = ( k > x k ≤ π h ( x ) = ( x k ≥ k < x ∈ \ CF K ( s + kµ ).2.3. The Ozsv´ath-Szab´o mapping cone.
Let λ be a longitude for K , so that µ · λ = 1.The mapping cone of Ozsv´ath and Szab´o [40] relates the Heegaard Floer homology of thefilling Y ( λ ) to the knot Floer homology of K . We recall its construction here.Since i h ( s − λ ) = i v ( s ), we have H ( \ CF K ( K, s − λ ) , d + d h ) ≃ d HF ( Y, i v ( s )) ≃ H ( \ CF K ( K, s ) , d + d v ) . This isomorphism is realized by a chain homotopy equivalence j : ( \ CF K ( K, s − λ ) , d + d h ) → ( \ CF K ( K, s ) , d + d v ) . (The map on homology induced by j is the canonical isomorphism of [27], although we willnot use this fact here.)For s ∈ Spin c ( Y, ∂Y ), let B K, s = ( \ CF K ( K, e s ) , d + d v ). We form two chain complexes A ( K ) = M s ∈ Spin c ( Y ) A K, s and B ( K ) = M s ∈ Spin c ( Y ) B K, s . There is a chain map f λ : A ( K ) → B ( K ) given by f = π v + j ◦ π h . (So if x ∈ A K, s , f λ ( x )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] ,Ozsv´ath and Szab´o prove Theorem 2.5. [40] d HF ( Y ( λ )) ≃ H ∗ ( X λ ( K )) . 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 X λ ( K, s ) = M n ∈ Z A K, s + nλ ⊕ M n ∈ Z B K, s + nλ . The summands are on one to one correspondence with elements of the quotient H ( Y ) / h λ i ≃ H ( Y ( λ )). The resulting decomposition on homology corresponds to the decomposition of d 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 com-plex X λ ( K ; F p ) = X λ ( K ) ⊗ F p . It follows from the universal coefficient theorem that d HF ( Y ( λ ); F p ) ≃ H ∗ ( X λ ( K ; F p )).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 underlyinggroup is A ( K ) ⊕ B ( K ), with differential given by dx = v ( x ) + h ( x ) for x ∈ A ( K ), dy = 0for y ∈ B ( K ). LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 9
Corollary 2.6. d HF ( Y ( λ ); F p ) ≃ H ( C λ ( K ; F p )) .Proof. The short exact sequence0 → B ( K ) ⊗ F p → X λ ( K ; F p ) → A ( K ) ⊗ F p → → B ( K ) → d HF ( Y ( λ ); F p ) → A ( K ) → B ( K ) → whose boundary map is given by v + h . An exact sequence over a field splits, so we get thestatement of the corollary. (cid:3) Splicing and surgery.
Suppose Y and Y are rational homology solid tori, and that ϕ : ∂Y → ∂Y is an orientation reversing diffeomorphism. The manifold Y ϕ = Y ∪ ϕ Y is obtained by splicing Y and Y together along ϕ . Choose a slope µ ∈ Sl ( ∂Y ), and let µ = ϕ ∗ ( µ ) be its image in Sl ( ∂Y ). Let Y i = Y i ( µ i ) be the corresponding Dehn fillings,and let K i = K µ i be their cores. Lemma 2.7. Y ϕ can be obtained by integral surgery on K K ⊂ Y Y . This is well-known, but an understanding of the proof will be useful in what follows, sowe sketch it here.
Proof.
Let Y ′ be the complement of K K . Y ′ is obtained by identifying an annulus ν ( µ ) ⊂ ∂Y with its image ν ( µ ) = ϕ ( ν ( µ )) ⊂ ∂Y . (Throughout the proof, we use thesame symbol to denote both a slope on the torus and a simple closed curve representing it.)Equivalently, Y ′ can be obtained by starting with the disjoint union of Y , Y and S × I × I and identifying S × I × ν ( µ ) and S × I × ν ( µ ). In this model, ∂Y ′ is aunion of four annuli: ∂Y − ν ( µ ), S × × I , ∂Y − ν ( µ ), and S × × I . The meridian µ of K K is homotopic to both µ and µ (and to the core of each of the four annuli.)Let λ be a longitude for µ , so that λ = − ϕ ( λ ) is a longitude for µ . We may assumethat λ ∩ ν ( µ ) = p × I ⊂ S × I ≃ ν ( µ ), and similarly for λ . Let λ ′ be the arc obtained byintersecting λ with ∂Y − ν ( µ ), and similarly for λ ′ . The union of the arcs λ ′ , p × × I, λ ′ , and p × × I is a longitude λ for K K . Attaching a 2-handle along λ is the same asattaching I × I × I to Y ′ , where the top and bottom edges I × / × I × / × λ ′ and λ ′ , and the sides 1 × / × I and 0 × / × I are identified withthe other arcs in λ . The resulting manifold can be obtained by starting with Y , Y andΣ × I , where Σ is a regular neighborhood of the 1-skeleton in T and identifying Σ × µ ∪ λ ⊂ ∂Y and Σ × ϕ . Finally, fillingin the spherical boundary component with B gives Y ∪ ( T × I ) ∪ Y = Y ϕ . (cid:3) From the proof, we see that H ( Y ′ ) ≃ H ( Y ) ⊕ H ( Y ) /R , where R is the subgroupgenerated by ( µ , µ ), and that under this isomorphism, λ = ( λ , ϕ ∗ ( λ )) = ( λ , − λ ).We make two remarks on the utility of this construction. First, it is quite flexible, in thesense that the choice of any meridian µ ∈ Sl ( ∂Y ) gives a different way of realizing thespliced 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 × D . Suppose h µ, λ i is our usual basis for H ( ∂Y ), and that h m, l i is the standard basis for H ( ∂S × D )(so l = [ ∂D ]). If we glue ∂Y to ∂ ( S × D ) in such a way that [ ∂D ] is identified with α = pµ + qλ ∈ H ( ∂Y ), then it is easy to see that µ is identified with − qm + p ∗ l , where pp ∗ ≡ q . Applying the lemma, we see that Y ( α ) is obtained by integer surgery on aknot 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 × D . (Inthe notation of [44], it is the simple knot K ( q, − p, χ ( \ HF K ( K ( q, − p, ∼ t q − t − . To use Lemma 2.7 to compute the Floer homology of a splice, we need to know how theknot Floer homology behaves under connected sum.
Lemma 2.8. [41] \ HF K ( K K ) ≃ \ HF K ( K ) ⊗ \ HF K ( K ) . The isomorphism is well-behaved with respect to Spin c structures, in the sense that χ ( \ HF K ( K K )) ∼ χ ( \ HF K ( K )) χ ( \ HF K ( K )) . It is also respects the differentials, in the sense that \ CF K ( K K , d + d v ) is homotopyequivalent to \ CF K ( K , d + d v ) ⊗ \ CF K ( K , d + d v ), and similarly for d h .In [41], Ozsv´ath and Szab´o combined the observations above with their mapping cone forinteger surgeries to express the Floer homology of any rational surgery as a mapping cone.3. Floer Simple Manifolds
In this section we use Ozsv´ath and Szab´o’s mapping cone formula to prove Proposition 1.3and to derive some basic facts about Floer simple manifolds. For the most part, these arestraightforward extensions of results in [39],[44], and [4]. We conclude by explaining how tocompute the bordered Floer homology of a Floer simple manifold Y in terms of τ ( Y ) and aFloer 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 thatsome nontrivial surgery on Y is also an L-space. Definition 3.1.
We say that \ HF K ( K, e s ) is a positive chain if it is generated by elements x , . . . , x n , y , . . . , y n − and the induced differentials ˜ d h , ˜ d v satisfy ˜ d v ( y i ) = ± x i +1 , ˜ d h ( y i ) = ± x i +1 , and ˜ d v ( x i ) = ˜ d h ( x i ) = 0 for all i . More generally, we say that \ HF K ( K ) consists ofpositive chains if \ CF K ( K, e s ) is a positive chain for each s ∈ Spin c ( Y ) , and that \ HF K ( K ) consists of coherent chains if either \ HF K ( K ) or \ HF K ( − K ) consists of positive chains,where − K ⊂ − Y is the mirror knot. Note that all the x i ’s in the definition must have the same relative Z / 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.Ozsv´ath and Szab´o proved in [39] that if K ⊂ S has an L-space surgery with positiveslope, then \ HF K ( K ) is a positive chain. The following generalization is an easy consequenceof 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 isalso 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], thebent group A K, s ≃ Z for all s ∈ Spin c ( Y, ∂Y ). The proof of Theorem 1.2 of [39] carries overunchanged to show that \ HF K ( K, e 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 LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 11 obtained by positive surgery on − K ⊂ − Y . − Y ′ is also an L-space, so \ HF K ( − K ) consistsof 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 composedof coherent chains. (cid:3) Lemma 3.3. If \ HF K ( K ) consists of coherent chains, then τ ( Y ) = X h ∈ S [ τ ( Y )] [ h ] . Proof.
We have τ ( Y ) ∼ χ ( \ HF K ( K ))(1 − [ µ ]) = X s ∈ M χ ( \ HF K ( K, e s )) s ! ∞ X i =0 [ µ ] i ! where M ⊂ Spin c ( Y, ∂Y ) is a set of coset representatives for the action of h µ i and χ ( \ HF K ( K, e s )) = X j ∈ Z χ ( \ HF K ( K, s + jµ ))[ µ ] j . The hypothesis that \ HF K ( K ) consists of coherent chains implies that the nonzero coeffi-cients of χ ( \ HF K ( K, e s )) alternate between +1 and −
1, and that the outermost coefficientsare +1. It follows that the coefficients of the product χ ( \ HF K ( K, e s )) (cid:0)P ∞ i =0 [ µ ] i (cid:1) are alleither 0 or +1, and hence that all the coefficients of τ ( Y ) are either 0 or 1 as well. (cid:3) Corollary 3.4.
Suppose \ HF K ( K ) is composed of coherent chains, and that φ ( µ ) > k Y k .Then K is Floer simple.Proof. By hypothesis, \ HF K ( K ) is composed of coherent chains, so to prove that K is Floersimple, it suffices to show that every monomial in χ ( \ HF K ( K )) appears with a positivecoefficient. As usual, we normalize τ ( Y ) = P h a h [ h ] so that a h = 0 whenever φ ( h ) <
0, and a = 0. We have χ ( \ HF K ( K )) ∼ (1 − [ µ ]) τ ( Y ) , so the coefficient of [ h ] in χ ( \ HF K ( K )) is a h − a h − µ . 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 ) > k Y k , then a h = 1 by Proposition 2.2. In either case, we see that thecoefficient of [ h ] in χ ( \ HF K ( K )) is either 0 or 1. (cid:3) Lemma 3.5. If \ HF K ( Y ) is composed of positive chains, there is an interval in Sl ( Y ) whose left endpoint is µ and which is contained in L ( Y ) .Proof. Since \ HF K ( K ) is composed of positive chains, the homology of each of its bentcomplexes is Z . Since the homology of the bent complexes computes d HF ( Y ( N µ + λ )) forsome 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 ). (cid:3) By considering mirrors, we see that if \ HF K ( K ) is composed of negative chains, then µ isthe right endpoint of a closed interval in L ( Y ). It follows that if K is Floer simple, then itis an interior point of an interval in L ( Y ). Conversely, if \ HF K ( K ) is composed of negativechains 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 chainsbut is not Floer simple, µ is in not in the interior of L ( Y ). • • • • • • • • • • • ⋆ ⋆ • ⋆ ◦ • ⋆ ◦ • ◦ ◦ Figure 1.
Part of a typical complex C L . Blue dots are shown by stars;red dots by hollow circles. Summands of each of the possible forms arevisible. Proof. (Of Proposition 1.3) If Y is Floer simple, then it has some filling Y ( α ) for which K α is Floer simple. As we observed above, α is contained in the interior of an interval in L ( Y ),so clearly |L ( Y ) | >
1. Conversely, if L ( Y ) >
1, then \ HF K ( K ) is composed of coherentchains, so L ( Y ) contains an interval. Now any interval in Sl ( Y ) contains elements α with φ ( α ) arbitrarily large. (To see this, identify Sl ( Y ) with Q using the canonical meridian andlongitude. If α a/b under this identification, then φ ( α ) = ka for some fixed k > K α ⊂ Y ( α ) is Floer simple, so Y is Floer simple. (cid:3) 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 L-space. To do so, we consider the set S black = S [ \ HF K ( K )] ⊂ Spin c ( Y, ∂Y ). Since K is Floersimple, S black is a set of coset representatives for the action of the subgroup h µ i ⊂ H ( Y ).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 < K with slope λ , where µ · λ = 1. We divide Spin c ( Y, ∂Y )into cosets for the action of h λ i . Each coset L is an affine copy of Z , so it has a natural order-ing. Each element of L is colored either black, red, or blue; elements which are sufficientlynegative 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. Proposition 3.6. Y ( λ ) is an L-space if and only if every coset for the action of h λ i isproperly colored.Proof. The argument is the same as the proof of Lemma 4.8 in [44]; we sketch it brieflyhere. We fix a prime p and use the mapping cone to compute d HF ( Y ( λ ); F p ). The mappingcone 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 isomorphicto 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 toprow represents A K, s + nλ ≃ F p , each dot in the bottom row represents B K, s + nλ ≃ F p , andthe arrows represent nonzero differentials. The chain of differentials breaks each time weencounter a red or blue dot, thus decomposing C L into smaller summands. Summandscorresponding 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 ineven Z / Z / d 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
LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 13 proposition follows from the fact that Y ( λ ) is an L-space if and only it is an F p L-space forevery prime p . (cid:3) Bordered Floer homology of Floer simple manifolds.
In this section, we showthat the bordered Floer homology [29] of a Floer simple manifold Y is determined by theTuraev torsion of Y together with a slope in the interior of L ( Y ). We very briefly reviewsome facts about bordered Floer homology; for more details see [29, 30].A bordered three-manifold is an oriented three-manifold Y equipped with a parametriza-tion (that is, a minimal handle decomposition) of its boundary. We will restrict our attentionto the case where ∂Y = T , in which case a parametrization is specified by a choice of twosimple closed curves µ, λ ∈ H ( ∂Y ) which satisfy µ · λ = 1.The type D bordered Floer homology \ CF D ( Y, µ, λ ) is a differential graded moduleover a certain F –algebra A ( Z ) associated to the torus. A ( Z ) is generated by elements ρ , ρ , ρ , ρ , ρ and ρ corresponding to certain arcs on the boundary of the 0-handlein the handle decomposition of ∂Y , together with a pair of idempotents ι , ι . FollowingChapter 11 of [29], we can think of the module structure as being specified by a pair ofvector spaces V , V over the field of two elements F , together with linear maps D , D , D : V → V D : V → V D : V → V D : V → V where \ CF D ( Y, µ, λ ) = A ( Z ) ⊗ F ( V ⊕ V ) and for x ∈ V ⊕ V , the differential is givenby ∂x = P ρ I D I ( x ) . In writing the above, we have assumed that \ CF D ( Y, µ, λ ) has been reduced with respectto all provincial differentials, so that V ≃ SF H ( Y, γ µ ) ≃ \ HF K ( K µ ) V ≃ \ HF K ( Y, γ λ ) ≃ HF K ( K λ )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 / \ CF D ( Y, µ, λ ) can be given a Z / Lemma 3.7.
The maps D and D preserve the homological Z / grading. If D hasparity i with respect to the Z / grading, then D , D and D have parity i , i and i ,respectively.Proof. We first consider the absolute grading on A ( Z ). By definition, algebra generatorscorresponding to arcs joining two ends of the same α arc have grading 1. (See definition 11 of[42] and the equations just preceding it.) In our case, this says that gr ρ ≡ gr ρ ≡
1. Fromthe relations ρ · ρ = ρ , ρ · ρ = ρ , and ρ · ρ = ρ , we see that gr ρ ≡ gr ρ + 1,gr ρ ≡ gr ρ + 1, and gr ρ ≡ gr ρ + 1 ≡ gr ρ . The statement now follows from the factthat gr ∂ x ≡ gr x + 1. (cid:3) We will also need to know how the D I ’s behave with respect to the Spin c grading. Letus write V s := \ HF K ( K µ , s ), so we have a decomposition V ≃ ⊕ s V s , and similarly for V ,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 fieldson Y with fixed behavior on ∂Y . (Recall that two nonvanishing vector fields are said to behomologous 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 aredifferent. Lemma 3.8.
There is a bijection j : Spin c ( Y, γ µ ) → Spin c ( Y, γ λ ) which respects the actionof H ( Y ) and for which D : V s → V j ( s ) D : V j ( s ) → V s − λ D : V s → V j ( s )+ λ + µ D : V s → V s − λ D : V j ( s ) → V j ( s )+ µ D : V s → V j ( s )+ µ 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 gradinglives in a set S ( H ) of homotopy classes of nonvanishing vector fields on Y which satisfycertain 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 beequal to v ι for some elementary idempotent ι .Similarly, Huang and Ramos consider the set G ( Z ) of homotopy classes of nonvanishingvector fields on ∂Y × [0 , v | ∂Y × = v ι and v | ∂Y × = v ι ′ for some elementary idempotents ι and ι ′ . They show that G ( Z ) forms a groupoid underconcatenation, and that it acts on the grading set S ( H ), again by concatenation. In section2.3 of [23], they construct explicit vector fields v I on ∂Y × [0 ,
1] associated to each ρ I ; thegrading 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 whichtakes 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 − · gr x , where λ is a vector field on ∂Y × [0 ,
1] which issupported in a ball. It follows that s ( ∂ x ) = s ( x ), and hence that p ( v I ) · s ( D I x ) = s ( x ).If s ∈ Spin c ( Y, γ µ ), we define j ( s ) = p ( v − ) · s . By construction, D : V s → V j ( s ) . Thefact that G ( Z ) is a groupoid implies that j is a bijection; j is equivariant with respect tothe action of H ( Y ) since we can arrange this action to take place in the interior of Y , awayfrom the region in which the concatenation takes place. Similarly, we see that s ( D x ) = p ( v − ) · s ( x ) = p ( v − · v ) · j ( s ( x )) . The set of homology classes of nonvanishing vector fields on ∂Y × [0 ,
1] which restrict to v ι on one end and v ι on the other is an affine copy of H ( ∂Y × [0 , ≃ H ( ∂Y ). Thusif I is the idempotent of the groupoid G ( Z ) corresponding to the idempotent ι , we musthave p ( v − · v ) = p ( I ) + α , for some α ∈ H ( ∂Y ). It follows that s ( D ( x )) = j ( s ( x )) + α for some universal element α ∈ H ( ∂Y ) which does not depend on Y or x . Comparing withLemma 11.42 of [29], we see that α = µ + λ . Thus D : V s → V j ( s )+ λ + µ as desired. Thearguments for the other D I ’s are very similar. (cid:3) Proposition 3.9.
Suppose that Y is Floer simple, that α ∈ Sl ( Y ) is a Floer simple fillingslope, and that µ, λ ∈ H ( ∂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 ofbasis bimodules in [30]. LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 15 p qr τ m ρ ⊗ ρ + ρ ⊗ ρ + ρ ⊗ ( ρ , ρ ) ρ ⊗ ρ + ρ ⊗ ( ρ , ρ ) ρ ⊗ ρ ρ ⊗ ρ ρ ⊗ ⊗ ρ q ps τ − l ρ ⊗ ρ ⊗ ρ ⊗ ρ ρ ⊗ ρ + ρ ⊗ ρ + ρ ⊗ ( ρ , ρ ) ρ ⊗ ( ρ , ρ ) + ρ ⊗ ρ ρ ⊗ ρ Figure 2.
Change of framing bimodules for the torus, taken from figureA.3 of [29].We choose µ to be a slope in the interior of L ( Y ) such that φ ( µ ) > k Y k , and take λ = λ − N µ , where λ is some class with µ · λ = 1, and N ≫
0. (We will specify belowhow large N needs to be.)The knots K µ , K λ are Floer simple, so all the elements of V have the same Z / V have the same Z / D = D =0 or D = D = 0. To see which of these two options hold, we consider the effect of a Dehntwist along µ . We have \ CF D ( Y, µ, λ + µ ) = \ CF DA ( τ µ ) ⊠ \ CF D ( Y, µ, λ )where the change of framing bimodule \ CF DA ( τ µ ) is shown in Figure 2.Writing \ CF D ( Y, µ, λ + µ ) = W ⊕ W , we have W = r ⊠ V ⊕ q ⊠ V . Denoteby D : W → W the contribution to ∂ coming from provincial differentials; then wehave H ( W , D ) = \ HF K ( K µ + λ ). By choosing N sufficiently large, we can ensure that µ + λ = λ − ( N − µ is in the interior of L ( Y ). It follows that \ HF K ( K µ + λ ) is Floersimple and has dimension equal to | H ( Y ( µ + λ )) | = | H ( Y λ ) | − | H ( Y µ ) | = dim V − dim V . Referring to the figure, we see that the only contribution to the provincial differential D comes from the arrow labeled 1 ⊗ ρ . Thus the map ρ : V → V is an injection. Similarly,by considering \ CF D ( Y, µ + λ, λ ) = \ CF DA ( τ − λ ) ⊠ \ CF D ( Y, µ, λ )we deduce that the map D : V → V is injective. Since D and D are nontrivial, we musthave D = D = 0.Let s max ∈ S [ \ HF K ( K µ )] be maximal, in the sense that if s max + α ∈ S [ \ HF K ( K µ )](where α ∈ H ( Y )), then φ ( α ) ≤ Lemma 3.10. j ( s max ) is maximal in S [ \ HF K ( K λ )] .Proof. It is well known [35] that \ HF K detects the Thurston norm, in the sense that if K ⊂ Y ( α ), then max { φ ( s − s ′ ) | s , s ′ ∈ S [ \ HF K ( K )] } = k Y k + | φ ( α )) | . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦• • • • • Figure 3.
Generators of \ CF D ( Y, m − l, − m + 2 l ), where Y is thecomplement of the negative trefoil in S . Dots in the top row representgenerators of V , dots in the bottom row generators of V . The horizontalposition of each generator indicates its Spin c grading. Potential componentsof the differential are shown by arrows: red (sloping right) for D , blue(sloping left) for D , and black (the arcs) for D .Choose nonzero elements x ∈ V s max , y ∈ V s min , where φ ( s max − s min ) = k Y k + φ ( µ ). Since D and D are injective, j ( s max ) and j ( s min )+ λ + µ are both in S [ \ HF K ( K λ )]. We compute φ ( j ( s max ) − ( j ( s min ) + µ + λ )) = k Y k + | φ ( λ ) | = max { φ ( s − s ′ ) | s , s ′ ∈ S [ \ HF K ( K λ )] } . It follows that j ( s max ) must be maximal and j ( s min + µ + λ ) must be minimal. (cid:3) We represent \ CF D ( Y, µ, λ ) by a directed graph like that shown in Figure 3, with a vertexfor each generator and an edge for each potential component of the differential; that is, foreach pair of generators x , y whose Z / 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 . We have already seen that D and D are both injective, so x is the starting point of one arrow labeled with D and one arrowlabeled with D . D = D = 0, so the only other possible arrows adjacent to x arelabeled by D . Now D shifts the Spin c grading by − λ , and φ ( − λ ) = N φ ( µ ) − φ ( λ ). Let S = S [ \ HF K ( K µ )]. We choose N sufficiently large that | φ ( λ ) | > max s ∈ S φ ( s ) − min s ∈ S φ ( s );then D vanishes for grading reasons.Next, if x is a generator of V , it can be a terminal point of an arrow labeled D or D , and either an initial or a terminal point of a arrow labeled D . We claim that x is aterminal point of an arrow of type D if and only if it is not an initial point of an arrow oftype D . 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 ( Y ). We consider the situation case by case,depending on the value of φ ( α ).(1) φ ( α ) <
0. In this case j ( s ) is unoccupied, and there is nothing to check. LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 17 (2) 0 ≤ φ ( α ) < φ ( µ ). In this region, χ ( \ HF K ( K µ )) and χ ( \ HF K ( K λ )) are both givenby τ ( Y ), so s is occupied if and only if j ( s ) is occupied. φ ( − α + µ ) >
0, so j ( s ) + µ is unoccupied.(3) φ ( µ ) ≤ φ ( α ) ≤ φ ( µ ) + k Y k . In this region j ( s ) is always occupied (see the argumentfor region 4) below), while s is occupied if and only if s + µ is not occupied. j ( s ) + µ is in region 2), so s is occupied if and only if j ( s ) + µ is not occupied.(4) φ ( µ ) + k Y k < φ ( α ) < | φ ( λ ) | . In this region s is unoccupied, while χ ( \ HF K ( K λ )) isgiven by τ ( Y ). By Proposition 2.2, both j ( s ) and j ( s ) + µ are always occupied.(5) | φ ( λ ) | < φ ( α ) < | φ ( λ ) | + k Y k . In this region, s is unoccupied. Since φ ( µ ) > | Y | , j ( s ) + µ is in region 4) and is always occupied.(6) | φ ( λ ) | + k Y k ≤ α . In this region, j ( s ) is unoccupied.This proves the claim. A very similar argument shows that x is a terminal point of anarrow of type D if and only if it is not the terminal point of an arrow of type D . Thestatement of the lemma follows. (cid:3) Since K µ and K λ are Floer simple, each arrow in the diagram corresponds to a map F → F . To determine the corresponding component of the differential, it suffices to knowwhether or not this map is 0. We will show that every map corresponding to an arrow inthe diagram is nonzero, thus completing the proof of Proposition 3.9. The maps D and D are injective, so any arrow labeled by D or D is nonzero. For the arrows labeled by D ,we argue as in the proof of Theorem 11.36 in [29]. By Proposition 11.30 of [29], there aremaps D , D , D , D , and D satisfying D ◦ D + D ◦ D + D ◦ D = 1 V D ◦ D + D ◦ D + D ◦ D = 1 V Since D = D = 0, it follows that if x is not in the image of D , it must be in theimage of D , and if x is not in the image of D , D ( x ) = 0. Comparing with the proof ofLemma 3.11, we see that every arrow in the diagram must correspond to a nonzero map. (cid:3) Intervals of L-space filling slopes
Now that the “proper coloring” condition of Proposition 3.6 is in place, we are equippedto tackle the problem of describing L-space intervals in terms of D τ ( Y ) and a slope from theinterior 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-manifoldwith 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 ( ∂Y ) / {± } , or equiva-lently, to elements of P ( H ( ∂Y )). Any choice of basis ( m, l ) for H ( ∂Y ) specifies homoge-neous coordinates nm + n ′ l [ n : n ′ ] on P ( H ( ∂Y )), to which we usually refer in terms ofthe affinization H ( ∂Y ) \ { } → Q ∪ {∞} , (1) nm + n ′ l n/n ′ . Let ι : H ( ∂Y ) → H ( Y ) be the map induced by inclusion. We fix a basis ( m, l ) for H ( ∂Y ) such that l is a generator of ker ι and m · l = 1. The generator l is the homologicallongitude of Y ; it is well defined up to sign. In contrast, the choice of m is only well definedup to the addition of a multiple of l . Consequently, the numerator of π ( nm + n ′ l ) = n/n ′ iscanonical (up to sign), but the denominator depends on the choice of m . To Dehn fill Y along a slope µ = nm + n ′ l ∈ H ( ∂Y ), one attaches a 2-handle alongthe simple closed curve associated to µ , and then fills in the remaining S boundary witha 3-ball. The resulting manifold, which we denote by Y ( µ ) or Y ( n/n ′ ), has homology H ( Y ( µ )) = H ( Y ) / ( ι ( µ )), which has order | n | if H ( Y ) is torsion free.Any non-zero Dehn filling Y ( µ l ) produces a knot K µ l := core( Y ( µ l ) \ Y ) ⊂ Y ( µ l ), onwhich one can now perform Dehn surgery. Whereas our conventionial choice of basis forDehn 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 involvesa canonical meridian , namely µ l , for the knot K µ l ⊂ Y ( µ l ), with the longitude λ l ∈ H ( ∂Y )(satisfying µ l · λ l = 1) only determined up to the addition of copies of µ l .Thus, for an arbitrary slope, say(2) µ = nm + n ′ l = αµ l + βλ l ∈ H ( ∂Y ) , 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 µ l (with respect to the basis ( µ l , λ l )). Notethat each of these conventional descriptions involves either a denominator or a numera-tor which is non-canonical. To dodge this problem, we can instead divide the canonicalnumerator of n/n ′ by the canonical denominator of α/β to obtain n/β , with(3) n := µ · l, β := µ l · µ = pn ′ − qn (where µ l = pm + ql ) , and with | n | = | H ( Y ( µ )) | when H ( Y ) is torsion free. Note that n/β is not a slope inthe conventional sense, since µ = n ( µ l /p ) + β ( l/p ), with µ l /p, l/p / ∈ H ( ∂Y ; Z ), and theprojective linear map P ( H ( ∂Y )) → P ( Z ), [ n : n ′ ] [ n : β ] is not surjective, havingdeterminant p . Still, since this map is injective, it is sufficient for cataloguing slopes. Infact, 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 (orjust surgery label ) of µ . Since(4) nn ′ = pq + 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 ( Y ), relative to the map ι : H ( ∂Y ) → H ( Y ), restrictingto the case of b ( Y ) = 1. The Universal Coefficients Theorem implies coker ι ∼ = H ( Y ) ∼ =Tors( H ( Y )). Thus, setting T := Tors( H ( Y )) and T ∂ := h ι ( l ) i = T ∩ ι ( H ( ∂Y )), we havecoker ι = H ( Y ) / (cid:0) h ι ( m ) i ⊕ T ∂ (cid:1) ∼ = T , which implies(5) ( H ( Y ) /T ) /ι ( m ) ∼ = T ∂ ∼ = Z /g, where g := | T ∂ | . In other words, any generator ¯ m for H ( Y ) /T will satisfy ι ( m ) ∈ ± g ¯ m + T .We shall always choose ¯ m so that ι ( m ) ∈ + g ¯ m + T .4.2. Conventions for Turaev torsion and D τ ( Y ). Recall our definition for D τ ( Y ) ⊂ H ( Y ) as the finite set(6) D τ ( Y ) := { x − y | x / ∈ S [ τ ( Y )] , y ∈ S [ τ ( Y )] } ∩ ι ( m Z ≥ + l Z ) , where τ ( Y ) is the Turaev torsion of Y , which we always normalize so that(7) 0 ∈ S [ τ ( Y )] , τ ( Y ) ∈ Z [[ t ]][ T ] , with t := [ ¯ m ] for any generator ¯ m of H ( Y ) /T ∼ = Z satisfying ι ( m ) ∈ ¯ m Z > + T . LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 19
When Y is Floer simple, we can also define the torsion complement ,(8) τ c ( Y ) := 11 − t X h ∈ T [ h ] − τ ( Y ) , with the Floer simplicity of Y guaranteeing that(9) S [ τ c ( Y )] = ¯ m Z ≥ ⊕ T \ S [ τ ( Y )] , so that D τ ( Y ) admits the alternative definition(10) D τ ( Y ) := ( S [ τ c ( Y )] − S [ τ ( Y )]) ∩ ι ( m Z ≥ + l Z ) . We shall often want to restrict our attention to the non-torsion elements of D τ ( Y ),(11) D τ > ( Y ) := ( S [ τ c ( Y )] − S [ τ ( Y )]) ∩ ι ( m Z > + l Z ) = D τ ( Y ) \ T, When we wish to emphasize our inclusion of the torsion elements of D τ ( Y ), we shall write D τ ≥ ( Y ) for 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. Proposition 4.1. If Y is Floer-simple, then the complement Γ( Y ) := ι ( m Z ≥ + l Z ) \ D τ ( Y ) is closed under addition.Proof. Suppose there exist x, y ∈ Γ( Y ) with x + y ∈ D τ ( Y ). Since x + y ∈ D τ ( Y ), weknow there exists z ∈ S [ τ ( Y )] for which x + y + z ∈ S [ τ c ( Y )]. If z + x ∈ S [ τ c ( Y )] then x = ( x + z ) − z ∈ D τ ( Y ), a contradiction. On the other hand, if s + x ∈ S [ τ ( Y )], then y = ( x + y + z ) − ( x + z ) ∈ D τ ( Y ), another contradiction. Thus x + y / ∈ D τ ( Y ). (cid:3) 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.4.3. Notation: Truncation and remainders.
Lastly, we need some basic arithmeticnotation. Henceforth in this paper, we use the conventional truncations ⌊·⌋ , ⌈·⌉ : Q → Z ,(12) ⌊ r ⌋ := max { z ∈ Z | z ≤ r } , ⌈ r ⌉ := min { z ∈ Z | z ≥ r } , and the less conventional notation [ · ] p : Z → { , . . . , | p |− } to select a representative modulo p , by projecting an integer to Z / | p | Z and then selecting its preimage in { , . . . , | p | − } ⊂ Z .In terms of our truncation notation,(13) [ a ] b = a − j ab k b, [ − a ] b = − a + l ab m b, when b > . Restating Theorem 1.6 as Theorem 4.2.
We are now equipped to re–expressTheorem 1.6 in a more practical form, describing the L-space slope interval in terms of anygiven 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 labelof the canonical longitude—we can always describe the interval of L-space surgery labels interms of its minimum and maximum in Q .That is, given an L-space slope µ l = pm + ql ∈ H ( ∂Y ) from the interior of the L-spaceinterval, Theorem 1.6 tells us that a Dehn filling Y ( µ ) is an L-space if and only if(14) π (˜ δ − ) ≤ π ( µ ) ≤ π (˜ δ + ) for all δ ∈ D τ > ( Y ) , where π denotes the surgery µ l -label,(15) π : H ( ∂Y ) \ { } → Q ∪ {∞} , µ π ( µ ) := ( µ l · µ ) / ( µ · l ) , and where, for each δ ∈ D τ > ( Y ), the lifts ˜ δ − , ˜ δ + ∈ ι − ( δ ), with π (˜ δ − ) < π (˜ δ + ), are the twolifts of δ closest to µ l with respect to π , assuming D τ > ( Y ) nonempty.Since D τ > ( Y ) ⊂ ι ( H ( ∂Y )), we can express any δ ∈ D τ > ( Y ) as δ = δι ( m ) + γι ( l ). Any lift˜ δ ∈ ι − ( δ ) of δ then takes the form ˜ δ = δm + ˜ γl , satisfying π (˜ δ ) = ( µ l · ˜ δ ) /δ = ( p ˜ γ − qδ ) /δ ,for some ˜ γ ≡ γ (mod g ). In other words, we have(16) n π (˜ δ ) (cid:12)(cid:12)(cid:12) ι (˜ δ ) = δ o = [ pγ − qδ ] pg + pg Z δ . Since π ( µ l ) = 0, the fact that µ l lies in the interior of the L-space interval implies that 0fails to belong to the above set, i.e. , that [ pγ − qδ ] pg = 0, for all δ ∈ D τ > ( Y ). The lifts ˜ δ + and ˜ δ − then evidently satisfy π (˜ δ + ) = [ pγ − qδ ] pg /δ and π (˜ δ − ) = ([ pγ − qδ ] pg − pg ) /δ forall δ ∈ D τ > , and we can rewrite Theorem 1.6 as follows. Theorem 4.2.
Suppose Y is Floer simple, with D τ > ( Y ) = ∅ . If µ l = pm + ql ∈ H ( ∂Y ) isan L-space slope for Y , satisfying b δ + := [ pγ − qδ ] pg = 0 for all δ = δι ( m ) + γι ( l ) ∈ D τ > ( Y ) ,then the Dehn filling Y ( µ ) is an L-space if and only if (17) b δ − δ ≤ µ l · µµ · l ≤ b δ + δ for all δ = δι ( m ) + γι ( l ) ∈ D τ > ( Y ) , where b δ − := b δ + − pg , and where we call ( µ l · µ ) / ( µ · l ) the surgery µ l -label for µ . If D τ > ( Y ) = ∅ ,then Y ( µ ) is an L-space if and only if µ / ∈ h l i , 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 − ( D τ ( Y )). This interval isopen if its endpoints are equal, and closed otherwise.Of course, one could express the above criterion in any other basis. To characterize L-space slopes in terms of conventional surgery coefficients, for surgery along the knot core K µ l ⊂ Y ( µ l ) \ Y associated to a given interior L-space slope µ l = pm + ql , one must firstchoose a longitude, say λ l := q ∗ m + p ∗ l , with µ l · λ l = 1 implying pp ∗ − qq ∗ = 1. Next, foreach δ ∈ D τ > ( Y ), we express the lifts ˜ δ + , ˜ δ − ∈ ι − ( δ ) flanking µ l as(18) ˜ δ + = a δ + µ l + b δ + λ l , ˜ δ − = a δ − µ l + b δ − λ l , with b δ + , b δ − a δ + , and a δ − satisfying(19) b δ + := [ pγ − qδ ] pg , b δ − := b δ + − pg, δ = a δ + p + b δ + q ∗ = a δ − p + b δ − q ∗ . When p >
0, a straightforward calculation shows that(20) a δ − b δ − < − q ∗ p < a δ + b δ + for all δ ∈ D τ > ( Y ) , and Theorem 4.2 takes the following form. Corollary 4.3.
Suppose Y is Floer simple, with D τ > ( Y ) = ∅ . If µ l = pm + ql with p > isan L-space slope for Y , satisfying b δ + := [ pγ − qδ ] pg = 0 for all δ = δι ( m ) + γι ( l ) ∈ D τ > ( Y ) ,then for any longitude λ l = q ∗ m + p ∗ l (with µ l · λ l = 1 ), the α/β surgery along K µ l ⊂ Y ( µ l ) —or equivalently, the Dehn filling Y ( µ ) with µ := αµ l + βλ l —is an L-space if and only if (21) αβ ≤ a δ − b δ − or a δ + b δ + ≤ αβ for all δ ∈ D τ > ( Y ) , where ι ( a δ + µ l + b δ + λ l ) = ι ( a δ − µ l + b δ − λ l ) = δ , with b δ − := b δ + − pg , for each δ ∈ D τ > ( Y ) . Insuch case, the left hand inequality obtains when β/n < , the right hand when β/n > , and LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 21 we regard both inequalities as vacuously true when β/n = 0 , where n := µ · l = αp + βq ∗ . If D τ > ( Y ) = ∅ , then Y ( µ ) 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 wetake µ l = pm + ql to be an interior L-space slope with p >
0, then for any δ = δι ( m )+ γι ( l ) ∈D τ > ( Y ), it follows from the two identities in (13) that(22) [ pγ − qδ ] pg = [ − qδ ] p + p h γ − l qp δ mi g ; − [ qδ − pγ ] pg = − [ qδ ] p − p hj qp δ k − γ i g , from which it follows that the lifts ˜ δ + , ˜ δ − ∈ ι − ( δ ) adjacent to µ l take the form(23) ˜ δ + = δm + (cid:24) qp δ (cid:25) + (cid:20) γ − (cid:24) qp δ (cid:25)(cid:21) g ! l, ˜ δ − = δm + (cid:22) qp δ (cid:23) − (cid:20)(cid:22) qp δ (cid:23) − γ (cid:21) g ! l, As expected, these are the lifts of δ with Dehn filling slope closest to p/q (regardless ofwhether p > Corollary 4.4.
Suppose Y is Floer simple, with D τ > ( Y ) = ∅ . If µ l = pm + ql is an L-space slope for Y , satisfying pγ − qδ pg ) for all δ = δι ( m ) + γι ( l ) ∈ D τ > ( Y ) , then µ = nm + n ′ l is an L-space slope for Y if and only if nn ′ ∈ I δ for all δ ∈ D τ > ( Y ) , where, foreach δ ∈ D τ > ( Y ) , I δ is the closed interval in Q ∪ {∞} which exludes 0 and has endpoints (24) δ l qp δ m + h γ − l qp δ mi g , δ j qp δ k − hj qp δ k − γ i g . If D τ > ( Y ) = ∅ , then Y ( µ ) is an L-space if and only if n = 0 . Example.
Suppose K ⊂ S is a Floer simple knot of positive genus g ( K ), with Alexanderpolynomial ∆( K ). Then Y := S \ ν ( K ) is Floer simple, and since K ⊂ S Floer simpleimplies deg ∆( K ) = 2 g ( K ), the hypothesis g ( K ) > D τ > ( Y ) = ∅ . Since H ( Y ) istorsion free, the endpoints of I δ reduce to δ/ l qp m and δ/ j qp k for each δ = δι ( m ) ∈ D τ > ( Y ).We already know that the infinity filling Y (1 m + 0 l ) = S is an L-space. Thus (if necessaryreplacing K with its mirror and using − nn ′ for nn ′ in (25)), we know that Y ( pm + 1 l ) is anL-space for any p > p > max δ ∈D τ> ( Y ) δ then makes the endpointsof each I δ become δ/ l p m = δ and δ/ j p k = + ∞ , and we recover the well known result thatfor n ′ = 0, Y ( nm + n ′ l ) is an L-space if and only if(25) nn ′ ≥ max δι ( m ) ∈D τ> ( Y ) δ = deg τ c ( Y ) = (deg ∆( K )) − g ( K ) − . Set-up for proof of Theorem 4.2.
We begin by making some simplifying assump-tions, 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 purposesof proving Theorem 4.2, we may assume, without loss of generality, that p, β > , n = 0 , pg > deg [ ¯ m ] τ c ( Y ) , and gcd( p, q ) = gcd( pg, β ) = 1 , where β := µ l · µ and g := |h ι ( l ) i| , with ι : H ( ∂Y ) → H ( Y ) the map induced on homology by inclusion. Proof.
Theorem 4.2 already correctly characterizes the cases of β = 0, corresponding to theDehn 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 anyL-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 > 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 . (cid:3) We henceforth consider the assumptions of Proposition 4.5 to hold. Given such initialdata, we have a primary tool from Heegaard Floer homology to determine whether µ is anL-space slope for the Floer simple manifold Y : namely, Proposition 3.6. To exploit thisproposition, 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 connectedsummed knot inside Y ( µ l ) L ( β, α ∗ ), for some α ∗ ≡ − α − (mod β ).4.6. Y ( µ ) as zero surgery on an L-space. To describe this construction more explicitly,we first let K u ⊂ S denote the unknot, and take ( m , l ) and ( m , l ) as respective basesfor H ( ∂Y ) and H ( ∂ ( S \ K u )), such that m · l = m · l = 1, with l generating ι − ( T ),where T := Tors( H ( Y ), and with l generating ker ι , where ι : H ( ∂Y ) → H ( Y ) and ι : H ( ∂ ( S \ K u )) → H ( S \ K u ) are the maps induced on homology by inclusion. Write(26) µ := nm + n ′ l , µ := µ l = pm + ql , µ := βm + α ∗ l for our test slope µ and given L-space slope µ = µ l , and for µ constructed to produce thedesired lens space ( S \ K u )( µ ) = L ( α ∗ , β ), with β := µ l · µ and α ∗ := [ − n − p ] β . Setting q ∗ := [ − q − ] p , write α , p ∗ , and β ∗ for the (integer) solutions to the respective equations n = αp + βq ∗ , pp ∗ − qq ∗ = 1, and ββ ∗ − α ∗ α = 1, so that(27) λ := q ∗ m + p ∗ l , λ := αm + β ∗ l serve as longitudes, satisfying µ · λ = µ · λ = 1. Note that this makes µ = αµ + βλ .Let Y denote the connected sum knot complement(28) Y := Y ( µ ) S \ K u )( µ ) \ K µ K µ , where K µ ⊂ Y ( µ ) and K µ ⊂ ( S \ K u )( µ ) = L ( β, α ∗ ) are the knot cores associated to therespective fillings by µ and µ . If we write ι : H ( ∂Y ) → H ( Y ), f : H ( Y ) → H ( Y ),and f : H ( S \ K u ) → H ( Y ) for the maps induced on homology by the correspondinginclusions, then f ⊕ f descends to the isomorphism,(29) (cid:0) H ( Y ) ⊕ H ( S \ K u ) (cid:1) / ( ι ( µ ) ∼ ι ( µ )) → H ( Y ) , which, since H ( S \ K u ) is torsion free, restricts to the isomorphism,(30) (cid:0) ι ( H ( ∂Y )) ⊕ ι ( H ( ∂ ( S \ K u ))) (cid:1) / ( ι ( µ ) ∼ ι ( µ )) → ι ( H ( ∂Y )) . For the knot K µ K µ with meridian µ , we can splice the longitudes λ and λ togetherto form a longitude of class λ ∈ ι − ( f ( λ )+ f ( λ )) ⊂ ι ( H ( ∂Y )). The Dehn filling Y ( λ )then has homology elements satisfying(31) f ι ( µ ) = f ι ( µ ) = βα f ι ( λ ) = − βα f ι ( λ ) , implying that f ι ( µ ) = f ι ( αµ + βλ ) = 0 in H ( Y ( λ )). Since, in addition, we knowthat Y is homeomorphic to Y , it follows that Y ( µ ) = Y ( λ ), and this is zero surgery onthe L-space Y ( µ ) L ( α ∗ , β ) = Y ( µ ). LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 23
Since gcd( pg, β ) = 1 and H ( S \ U ) is torsion free, it follows from the isomorphisms (29)and (30) that f restricts to isomorphisms T ∼ → T and T ∂ ∼ → T ∂ , where T := Tors( H ( Y )), T := Tors( H ( Y )), T ∂ := T ∩ ι ( H ( ∂Y )), and T ∂ := T ∩ ι ( H ( ∂Y )). It also follows thatwe can choose l ∈ ι − ( T ∂ ) and m ∈ H ( ∂Y ) with m · l = 1 such that f and f satisfy f : ι ( m ) βι ( m ) , f : ι ( m ) pι ( m ) + qξι ( l ) , (32) f : ι ( l ) βξι ( l ) , on the images of ι and ι , for some ξ ∈ Z /g , with g := (cid:12)(cid:12) T ∂ (cid:12)(cid:12) = (cid:12)(cid:12) T ∂ (cid:12)(cid:12) . We then have ι ( µ ) = f ι ( µ ) = f ι ( µ ) = βpι ( m ) + βqξι ( l ) , (33) ι ( λ ) = f ι ( λ ) + f ι ( λ ) = nι ( m ) + n ′ ξι ( l ) , where we used the facts that(34) n = αp + βq ∗ , n ′ = αq + βp ∗ . The condition that µ · λ = 1 determines ξ , which we shall not need.4.7. Applying the “coloring condition” of Proposition 3.6.
Since this section uses theEuler 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 introducegenerators ¯ m , ¯ m , and ¯ m for H ( Y ) /T , H ( Y ) /T , and H ( S \ K u ), respectively, withsigns chosen so that(35) ι ( m ) ∈ + g ¯ m + T, ι ( m ) ∈ + g ¯ m + T , ι ( m ) = ¯ m . For notational brevity, we also set(36) t := [ ¯ m ] , t := [ ¯ m ] , t := [ ¯ m ] , where [ · ] indicates inclusion into the Laurent series group ring of the relevant homologygroup.In order to use Proposition 3.6, we need the support of the Euler characteristic of theknot Floer homology of K ⊂ Y ( λ ). Since \ HF K tensors on connected sums, its Eulercharacteristic χ d hfk turns tensor product into multiplication, and the support function S [ · ]on (Laurent series) group rings converts this multiplication of polynomials into addition ofsets, yielding(37) S h χ d hfk ( Y ( λ ) , K ) i = f S h χ d hfk ( Y ( µ ) , K µ ) i + f S h χ d hfk (( S \ K u )( µ ) , K µ ) i . Proposition 2.1 tells us that S h χ d hfk ( Y ( µ ) , K µ ) i = S (cid:2) (1 − [ ι ( µ )]) · (cid:0) (1 − t ) − P h ∈ T [ h ] − τ c ( Y ) (cid:1)(cid:3) (38) = S (cid:20) − t pg − t P h ∈ T [ h ] − τ c ( Y ) + [ ι ( µ )] τ c ( Y ) (cid:21) = ( S [ τ ( Y )] ∩ ( { , . . . , pg − } ¯ m + T )) ∐ ( S [ τ c ( Y )] + ι ( µ ))) , where τ ( Y ) ∈ Z [ t − , t ]][ T ] ⊃ Z [ H ( Y )] is the Turaev torsion, τ c ( Y ) is the torsion com-plement as defined in (8), and we used our simplifying assumption that deg t τ c ( Y ) < pg .Similarly, we have S h χ d hfk (( S \ K u )( µ ) , K µ ) i = S (cid:2) (1 − [( ι ( µ )]) · τ ( S \ K u ) (cid:3) (39) = S [(1 − t β ) / (1 − t )]= { , . . . , β − } ι ( m ) . Thus, if we set A := f ( S [ τ ( Y )] ∩ ( { , . . . , pg − } ¯ m + T )) + { , . . . , β − } f ι ( m ) , (40) A := f S [ τ c ( Y )] + ι ( µ ) + { , . . . , β − } f ι ( m ) , then in the language of Proposition 3.6, we have S black := S h χ d hfk ( Y ( λ ) , K ) i = A ∐ A , (41) S red := S black + ι ( µ ) Z > , S blue := S black − ι ( µ ) Z > . Using the fact that ι ( µ ) = βf ι ( m ), one can easily verify that( S blue − S red ) ∩ ( ¯ m Z > + T ) = (( A − ι ( µ )) − ( A + ι ( µ ))) ∩ ( ¯ m Z > + T )(42) = ( f ( S [ τ c ( Y )] − S [ τ ( Y )]) − f ι ( m ) Z > ) ∩ ( ¯ m Z > + T ) . Proposition 3.6 then implies Y ( λ ) is an L-space if and only if(43) ι ( λ ) Z ∩ ( S blue − S red ) ∩ ( ¯ m Z > + T ) = ∅ . Suppose the above set is nonempty, hence contains some element bι ( λ ) such that(44) bι ( λ ) = f ( h c − h ) − kf ι ( m ) ∈ ¯ m Z > + T with b ∈ Z =0 , k ∈ Z > , h c ∈ S [ τ c ( Y )], and h ∈ S [ τ ( Y )]. Since bι ( λ ) , kf ι ( m ) ∈ ι ( H ( Y )), we know that f ( h c − h ) ∈ ι ( H ( Y )), implying h c − h ∈ ι ( H ( Y )). Moreover,since bι ( λ ) ∈ ¯ m Z > + T , we know that h c − h ∈ ¯ m Z > + T . In other words,(45) h c − h ∈ ( S [ τ c ( Y )] − S [ τ ( Y )]) ∩ ι ( m Z > + l Z ) =: D τ > ( Y ) . Writing h c − h = δι ( m ) + γι ( l ) ∈ D τ > ( Y ) and evaulating f , f , and ι ( λ ) as expressedin (32) and (33), we transform (44) into(46) ( bn ) ι ( m ) + ( bn ′ ) ξι ( l ) = ( βδ − kp ) ι ( m ) + ( βγ − kq ) ξι ( l ) , which, since nm + n ′ l = αµ + βλ = ( αp + βq ∗ ) m +( αq + βp ∗ ) l , yields the two equations b ( αp + βq ∗ ) = βδ − kp > , (47) b ( αq + βp ∗ ) ≡ βγ − kq (mod g ) . (48)One can use the identity pp ∗ − qq ∗ = 1 to solve the above two equations simultaneouslyfor 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 ).4.8. Completing the proof of Theorem 4.2.
For each δ = δι ( m ) + γι ( l ) ∈ D τ > ( Y ),set b δ − := [ pγ − qδ ] pg − pg and b δ + := [ pγ − qδ ] pg . Note that our earlier assumption of pg > deg t τ c ( Y ) ensures that b δ + = 0 and | b δ − | < pg . We want to show that Y ( λ µ ) is anL-space—or in other words, that (47) and (48) have no solution ( b, k ) ∈ Z × Z > for any δι ( m ) + γι ( l ) ∈ D τ > ( Y )—if and only if(49) b δ − δ ≤ βn ≤ b δ + δ for all δ = δι ( m ) + γι ( l ) ∈ D τ > ( Y ) . First, consider the case in which n >
0. Suppose there exists δ = δι ( m )+ γι ( l ) ∈ D τ > ( Y )for which β/n > b δ + /δ . Since n, δ >
0, this implies 0 < b δ + n < βδ . Thus, since b δ + n ≡ ( − qδ )( βq ∗ ) ≡ βδ (mod p ), there exists k ∈ Z > such that b δ + n = βδ − k p >
0. Thus ( b δ + , k )provides a solution for ( b, k ) in (47), which, together with the relation b δ + ≡ pγ − qδ (mod g ),implies (48) also holds for ( b, k ) = ( b δ + , k ), and so Y ( λ ) is not an L-space.Conversely, suppose that Y ( λ ) is not an L-space. Then there exist δ = δι ( m )+ γι ( l ) ∈D τ > ( Y ) and ( b, k ) ∈ Z × Z > for which (47) and (48) hold. In particular, (47) implies bn < βδ LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 25 and b >
0, while (47) and (48) together imply b ≡ b δ + (mod pg ), requiring b ≥ b δ + . Thus βδ > bn ≥ b δ + n , implying β/n > b δ + /δ .The argument for the case of n < (cid:3) Seifert Fibered L-spaces
To illustrate the usage of our new L-space interval tool D τ , in this section we exploitTheorem 4.2 to offer a simple alternative proof of a known result: namely, the classificationof Seifert fibered spaces over S which are L-spaces. We restrict to the S case because itis the most interesting one, as no higher genus Seifert fibered spaces are L-spaces, and alloriented Seifert fibered spaces over RP are L-spaces [7].5.1. Seifert fibered L-spaces, a history.
Up until now, the classification of Seifert fiberedL-spaces has relied, at least in one direction, on the classification of oriented Seifert fiberedspaces M over S admitting transverse foliations, a problem which dates back at least to1981, when Eisenbud, Hirsch, and Neumann [11] re-expressed this foliations problem interms of a criterion on representations of π ( M ) in ^ Homeo + S , the universal cover of thegroup of orientation-preserving homeomorphisms of S .A few years later, Jankins and Neumann [24] reformulated the criterion of [11] in termsof Poincar´e’s “rotation number” invariant on ^ Homeo + S , a development which, along withthe correct conjecture that this criterion is met in ^ Homeo + S if and only if it is met in asmooth Lie subgroup thereof, allowed them to write down an explicit characterization ofSeifert fibered manifolds over S admitting transverse foliations. With the exception of onespecial case, they also showed that this list was complete. It took more than a decade beforeNaimi [33] resolved this outstanding case using dynamical methods, and more than a decadeafter that before Calegari and Walker [9] generalized Naimi’s methods to provide a proof ofthe 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 weaklysymplectically fillable contact structure to any C cooriented taut foliation on a closedthree-manifold—a result which Kazez and Roberts [28], and independently Bowden [5],have recently extended to C foliations. Since Ozsv´ath and Szab´o have [38] shown that thiscontact structure gives rise to a nontrivial class in Heegaard Floer homology, this provesthat L-spaces do not admit co-oriented taut foliations.In the converse direction, Lisca and Mati´c [31] proved that a Seifert fibered manifold M over S admits contact structures in each orientation which are transverse to the fibrationif and only if M belongs to the explicit set characterized by Jankins and Neumann. Liscaand Stipsicz then showed [32] that if there is an orientation on a Seifert fibered manifold M over S for which no positive contact structure is transverse to the fibration, then M is anL-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 as a corollaryof Theorem 5.1 below. As for our L-space classification itself, however, the proof no longerrequires foliations, dynamical methods, or even (after the proof of Theorem 4.2) contactor symplectic geometry. It only uses ordinary homology and one computation of Turaevtorsion from a homology presentation. Conventions and bases.
To construct a Seifert-fibered space with n exceptionalfibers over S , we start with the trivial circle fibration S × S , and remove n + 1 solid tori, S × D i , i ∈ { , . . . , n } , yielding a trivial circle fibration over the n + 1–punctured sphere,(50) ˆ Y := S × ( S \ ` ni =0 D i ) , ∂ ˆ Y = ` ni =0 ∂ i ˆ Y , where ∂ i ˆ Y denotes the i th toroidal boundary component, ∂ i ˆ Y := − ∂ ( S × D i ).Next, we choose presentations for H ( ˆ Y ) and H ( ∂ i ˆ Y ) in terms of the regular fiber class f ∈ H ( ˆ Y ) and classes horizontal to this fiber. For each i ∈ { , . . . , n } , we take ( ˜ f i , − ˜ h i )as a reverse-oriented basis for H ( ∂ i ˆ Y ). Here, ˜ h i ∈ H ( ∂ i ˆ Y ) denotes the meridian of theexcised solid torus S × D i , and if we write ˆ ι i : H ( ∂ i ˆ Y ) → H ( ˆ Y ) for the map inducedby inclusion, then ˜ f i ∈ ˆ ι − i ( f ) denotes the lift of f satisfying ( ˜ f i · ˜ h i ) | ∂ i ˆ Y = 1. Settingeach h i := ˆ ι i (˜ h i ) ∈ H ( ˆ Y ), 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 . Infact, since any one of the h i can be regarded as the class of minus the boundary of thisdisk, with the remaining h i summing to a class equal to the boundary of the disk, we have P ni =0 h i = 0, so that H ( ˆ Y ) has presentation(51) H ( ˆ Y ) = h f, h , . . . , h n | P ni =0 h i = 0 i . To specify a Seifert fibered space, one simply lists the Dehn filling slopes, in terms ofthe basis ( ˜ f i , − ˜ h i ) for each H ( ∂ i ˆ Y ), of the n + 1 toroidal boundary components of ˆ Y ,conventionally filling ∂ Y with an integer slope and the remaining ∂ i Y with nonintegerslopes. That is, for any e , r , . . . , r n ∈ Z and s , . . . , s n ∈ Z =0 with each r i s i / ∈ Z , the Seifertfibered space M ( e ; r s , . . . , r n s n ) denotes the Dehn filling of ˆ Y along the slopes µ := e ˜ f − ˜ h , (52) µ i := r i ˜ f i − s i ˜ h i , i ∈ { , . . . , n } . The resulting manifold has first homology(53) H (cid:16) M ( e ; r s , . . . , r n s n ) (cid:17) = h f, h , . . . , h n | P ni =0 h i = ˆ ι ( µ ) = . . . = ˆ ι n ( µ n ) = 0 i . Note that for any ( z , . . . , z n ) ∈ Z n +1 satisfying P ni =0 z i = 0, the change of basis h i h i + z i f , i ∈ { , . . . , n } , yields the reparameterization(54) M ( e ; r s , . . . , r n s n ) M ( e + z ; r s + z , . . . , r n s n + z n ) . In addition, M ( e ; r s , . . . , r n s n ) admits an orientation reversing homeomorphism,(55) − M ( e ; r s , . . . , r n s n ) = M ( − e ; − r s , . . . , − r n s n ) . Statement of L-space classification.
We are now able to state our result.
Theorem 5.1. If M ( e ; r s , . . . , r n s n ) denotes a Seifert fibered space over S with n > exceptional fibers, then M ( e ; r s , . . . , r n s n ) is not an L-space if and only if e + P ni =1 r i s i = 0 or (56) − e + min Remark. If we take each s i > 0, then inequality (56) is equivalent to the condition that(57) min 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-fibercomplement in a Seifert fibered space. That is, for any j ∈ { , . . . , n } , the above theoremimplies that M ( e ; r s , . . . , r n s n ) is an L-space if and only if(58) − e x − − X i = j (cid:24) r i xs i (cid:25) ≥ (cid:24) r j xs j (cid:25) or − e x − X i = j (cid:22) r i xs i (cid:23) ≤ (cid:22) r j xs j (cid:23) for all x ∈ { , . . . , s − } . Since the above expressions are integers, (58) holds if and only if(59) − e x − − X i = j (cid:24) r i xs i (cid:25) ≥ r j xs j or − e x − X i = j (cid:22) r i xs i (cid:23) ≤ r j xs j . Dividing both sides by x then gives the following result. Corollary 5.2. If M ( e ; r s , . . . , r n s n ) , with each s i > , denotes a Seifert fibered space over S with n > exceptional fibers, then for any j ∈ { , . . . , n } , M ( e ; r s , . . . , r n s n ) is an L-spaceif and only if e + P ni =1 r i s i = 0 and (60) r j s j ≤ − e + min Webegin by expressing M ( e ; r s , . . . , r n s n ) as the Dehn filling of a Floer simple manifold Y . Fornow, we demand that 0 < r i < s i and gcd( r i , s i ) = 1 for each i ∈ { , . . . , n } . Let Y denotethe regular-fiber complement(61) Y := M (0; r s , . . . , r n s n ) \ ( S × D ) , so that Y ( µ ) = M ( e ; r s , . . . , r n s n ). Regarding Y as a partial Dehn filling of ˆ Y , we have(62) H ( Y ) = h f, h , . . . , h n | P ni =0 h i = ˆ ι ( µ ) = . . . = ˆ ι n ( µ n ) = 0 i . Writing ι : H ( ∂Y ) → H ( Y ) for the map induced by inclusion, and identifying ˜ h and˜ f with their respective images under the canonical isomorphism H ( ∂ ˆ Y ) → H ( ∂Y ), weagain have ι (˜ h ) = h and ι ( ˜ f ) = f , but in the sense of the above presentation for H ( Y ).Define(63) S gcd := gcd (cid:18) Q ni =1 s i s , . . . , Q ni =1 s i s n (cid:19) , s := Q ni =1 s i S gcd , noting that this makes s the least common multiple of s , . . . , s n . Note that if we set(64) l := p ˜ f + q ∗ ˜ h , with p := n X i =1 r i s i sg , q ∗ := sg , g := gcd n X i =1 r i s i s, s ! , then l is primitive in H ( ∂Y ). In addition, since h = − P ni =1 h i , we have(65) 0 = n X i =1 ss i ˆ ι i ( µ i ) = n X i =1 r i s i sf + sh = gι ( l ) . Thus ι ( l ) ∈ H ( Y ) is torsion, and so l is also a canonical longitude. Moreover, since gι ( l ) = P ni =1 ss i ˆ ι i ( µ i ) = 0 is a primitive linear combination of the relations in the presentation of H ( Y ) in (62), we have g = |h ι ( l ) i| . Choosing any m ∈ H ( ∂Y ) satisfying m · l = 1, andwriting m = − q ˜ f − p ∗ ˜ h , allows one to solve for ˜ f and ˜ h in terms of m and l .Now, since all r i s i > Y ( − h ) = M (0; r s , . . . , r n s n ) is an L-space, so we may take µ l := − ˜ h as our given L-spacefilling slope, and choose λ l = ˜ f for its longitude, with µ l · λ l = − ˜ h · ˜ f = 1. We then have(66) µ l := − ˜ h = pm + ql, λ l := ˜ f = q ∗ m + p ∗ l, with p and q ∗ as in (64), and with q and p ∗ solving the diophantine equation pp ∗ − qq ∗ = 1.5.5. Computation of D τ ( Y ) . To compute D τ ( Y ), we need the Turaev torsion, τ ( Y ).Recall that Y is a union along torus boundaries of trivial circle fibrations,(67) Y = S × ( S \ ` ni =0 D i ) ∪ S × D ∪ . . . ∪ S n × D n . The leftmost S above, corresponding to the regular fiber in ˆ Y , has class ˆ ι ( λ l ) = f ∈ H ( ˆ Y ).Similarly, for i ∈ { , . . . , n } , each S i above has class ˆ ι i ( λ i ) ∈ H ( ˆ Y ), where λ i is anylongitude satisfying ( µ i · λ i ) | ∂ i ˆ Y = 1. Since each ˆ ι i ( µ i ) = 0, each class ˆ ι i ( λ i ) is independentof the choice of λ i . The Turaev torsion then obeys a product rule for unions along torusboundaries [45], yielding τ ( Y ) := (1 − [ˆ ι ( λ l )]) − χ ( S \ ` ni =0 D i ) Q ni =1 (1 − [ˆ ι i ( λ i )]) − χ ( D i ) (68) := (1 − [ˆ ι ( λ l )]) n − Q ni =1 (1 − [ˆ ι i ( λ i )]) − , where [ · ] denotes inclusion of H ( ˆ Y ) into the Laurent series group ring for H ( ˆ Y ).These ˆ ι i ( λ i ) bear simple relationships to ι ( µ l ) and ι ( λ l ). That is, we claim that(69) ι ( µ l ) = P ni =1 r i ˆ ι i ( λ i ) , and ι ( λ l ) = s i ˆ ι i ( λ i ) for each i ∈ { , . . . , n } . To see this, note that since each µ i = r i ˜ f i − s i ˜ h i , with ( ˜ f i · ˜ h i ) | ∂ i ˆ Y = ( µ i · λ i ) | ∂ i ˆ Y = 1, weknow there exist r ∗ i , s ∗ i ∈ Z such that(70) ˆ ι i ( λ i ) = s ∗ i f + r ∗ i h i , r i r ∗ i + s i s ∗ i = 1 , implying that r i ˆ ι i ( λ i ) = s ∗ i ( r i f ) + r i r ∗ i h i = s ∗ i ( s i h i ) + r i r ∗ i h i = h i , (71) s i ˆ ι i ( λ i ) = s i s ∗ i f + r ∗ i ( s i h i ) = s i s ∗ i f + r ∗ i ( r i f ) = f = ι ( λ l ) . (72)Thus, since ι ( µ l ) = − h = P ni =1 h i , (69) holds in H ( Y ).Since ι ( λ l ) = s i ˆ ι i ( λ i ) for each i ∈ { , . . . , n } , we may rewrite τ ( Y ) as(73) τ ( Y ) = 11 − [ ι ( λ l )] n Y i =1 − [ˆ ι i ( λ i )] s i − [ˆ ι i ( λ i )] , which has support(74) S [ τ ( Y )] = { ι ( λ l ) Z ≥ } + { P ni =1 y i ˆ ι i ( λ i ) | y i ∈ { , . . . , s i − }} . LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 29 Since Y has multiple L-space fillings, it is Floer simple, and so each element of H ( Y ) hascoefficient 0 or 1 in τ ( Y ), and the torsion complement τ c ( Y ) has support(75) S [ τ c ( Y )] = {− jι ( λ l )+ P ni =1 y i ˆ ι i ( λ i ) | j ∈ { , . . . , n − } , y i ∈ { , . . . , s i − }}∩ H ( Y ) ≥ , where H ( Y ) ≥ := { w ∈ H ( Y ) | φ ( w ) ≥ } for any homomorphism φ : H ( Y ) → Z satifying φ ( ι ( m )) > s i ˆ ι i ( λ i ) = ι ( λ l ) for each i ∈ { , . . . , n } , it follows from (74) that S [ τ ( Y )] isadditively closed, which, in turn, implies that(76) ( S [ τ c ( Y )] − S [ τ ( Y )]) ∩ H ( Y ) ≥ = S [ τ c ( Y )] , so that D τ ( Y ) is the intersection D τ ( Y ) = S [ τ c ( Y )] ∩ ι ( H ( ∂Y )). By (69), we know that ι ( H ( ∂Y )) = Span { ι ( µ l ) , ι ( λ l ) } (77) = Span { ι ( µ l ) = P ni =1 r i ˆ ι i ( λ i ) , ι ( λ l ) = s ˆ ι ( λ ) = . . . = s n ˆ ι n ( λ n ) } . Now, for any j ∈ { , . . . , n − } and ( y , . . . , y n ) ∈ Q ni =1 { , . . . , s i − } , we have(78) − jι ( λ l ) + P ni =1 y i ˆ ι i ( λ i ) = P ni =1 ( y i + z i s i )ˆ ι i ( λ i ) − ( j + P ni =1 z i ) ι ( λ l )for any ( z , . . . , z n ) ∈ Z n . Thus, − jι ( λ l ) + P ni =1 y i ˆ ι i ( λ i ) ∈ ι ( H ( ∂Y )) if and only if thereexist ( z , . . . , z n ) ∈ Z n and x ∈ Z for which(79) ( y + z s , . . . , y n + z n s n ) = ( r x, . . . , r n x ) . In such case, we have y i = [ r i x ] s i and z i = j r i xs i k for each i ∈ { , . . . , n } .We can therefore parameterize D τ ( Y ) = S [ τ c ( Y )] ∩ ι ( H ( ∂Y )) as D τ ( Y ) = (cid:8) δ jx | j ∈ { , . . . , n − } , x ∈ { , . . . , s − } , δ jx ≥ (cid:9) , with(80) δ jx := a j − x ι ( µ l ) + b j − x ι ( λ l ) , a j − x := x, b j − x := − j − n X i =1 (cid:22) r i xs i (cid:23) ,δ jx := a j − x p + b j − x q ∗ = sg − j + n X i =1 [ r i x ] s i s i ! , where δ jx := ˜ δ jx · l for any ˜ δ jx ∈ ι − ( δ jx ). Since δ jx is invariant under the action x x + s ,it suffices to choose a fundamental domain of length s for x ∈ Z . The above expression for D τ ( Y ) uses the fundamental domain x ∈ { , . . . , s − } , but excludes 0, since δ j < j ∈ { , . . . , n − } .5.6. Application of Theorem 4.2/Corollary 4.3. This particular choice of fundamentaldomain ensures that for all δ jx ∈ D τ > ( Y ), we have b j − x = b δ jx − and a j − x = a δ jx − in the sense ofCorollary 4.3. That is, for all j ∈ { , . . . , n − } and x ∈ { , . . . , s − } with δ jx > 0, we have(81) 0 < − b j − x = a j − x pq ∗ − δ jx q ∗ = n X i =1 r i xs i − gs δ jx < n X i =1 r i ss i − pg. This makes a j − x µ l + b j − x λ l ∈ ι − ( δ jx ) one of the two lifts of δ jx closest to µ l in P ( H ( ∂Y )),and the closest lift of δ jx on the other side of µ l is a j + x µ l + b j + x λ l ∈ ι − ( δ jx ), where a j + x := a δ jx + = a j − x − q ∗ g = − ( s − x ) , (82) b j + x := b δ jx + = b j − x + pg = − j + n X i =1 (cid:24) r i ( s − x ) s i (cid:25) . To use Corollary 4.3 on M ( e ; r s , . . . , r n s n ) = Y ( µ ), we shall also want the ( µ l , λ l )-surgerycoefficients for µ , and the value of µ · l . Since µ = e ˜ f − ˜ h and l = p ˜ f + q ∗ ˜ h , with µ l = − ˜ h , λ l = ˜ f , p = sg P ni =1 r i s i , and q ∗ = sg , we have µ = αµ l + βλ l , α := 1 , β := e , (83) µ · l = e q ∗ + p = sg e + n X i =1 r i s i ! . Since Y ( µ ) is never an L-space when µ · l = 0, and since the case of e + P ni =1 r i xs i = 0 istreated separately in the theorem statement, we henceforth restrict to the case of µ · l = 0.Suppose that D τ > ( Y ) = ∅ . In this case, Corollary 4.3 tells us that M ( e ; r s , . . . , r n s n ) = Y ( µ ) is an L-space if and only if(84) αβ := 1 e ≤ xb j − x =: a j − x b j − x or a j + x b j + x := − ( s − x ) b j + x ≤ e =: αβ for all j ∈ { , . . . , n − } and x ∈ { , . . . , s − } with δ jx > 0, and moreover the left-hand(respectively right-hand) inequality obtains only if β/ ( µ · l ) < β/ ( µ · l ) > β = e < 0. Then M ( e ; r s , . . . , r n s n ) is an L-space if and only if(85) ≥ − e + b j − x /x for all j and x with δ jx > µ · l > ≤ − e − b j + x / ( s − x ) for all j and x with δ jx > µ · l < µ · l = 0 . Note that for all j ∈ { , . . . , n − } and x ∈ { , . . . , s − } , δ jx = a j ± x p + b j ± x q ∗ implies(86) b j − x /x = δ jx / ( q ∗ x ) − p/q ∗ , − b j + x / ( s − x ) = − δ jx / ( q ∗ ( s − x )) − p/q ∗ . Thus b j − x /x is never maximized and − b j + x / ( s − x ) is never minimized when δ x ≤ 0, so wecan remove the δ jx > b j − x /x is never maximized and − b j + x / ( s − x ) is never minimized when j > 1, so it suffices to fix j = 1. Reparameterizingthe second case of (85) by s − x x then transforms (85) into the condition(87) 0 ≤ − e + min 0, the right-hand inequality in (84) holds for all j ∈ { , . . . , n − } and x ∈ { , . . . , s − } . Accordingly, when e ≥ 0, (87) always holds (viaits right-hand inequality).Lastly, suppose that D τ > ( Y ) = ∅ . Since we have excluded the case of µ · l = 0, thisimplies that Y ( µ ) = M ( e ; r s , . . . , r n s n ) is an L-space, so we must show that (87) holds. Tosee this, first note that the negation of (87) is equivalent to the inequality(88) min The introduction to Section 5 discusses how, for Seifert fibered spaces over S (althoughthe same is true for all Seifert fibered spaces [7, 14]), the property of admitting a coorientedtaut 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 equivalentproperties belongs to a larger list. Theorem 6.1 ([11, 38, 32, 8]) . Suppose M is a Seifert fibered space over S . Then thefollowing are equivalent: (1) M admits a cooriented taut foliation. (2. ρ ) There exists a homomorphism ρ : π ( M ) → Homeo + R with non-trivial image. (2.LO) The fundamental group π ( M ) admits a left ordering. (3) M is not an L-space.Summary of Proof. Our idiosyncratic numbering owes to a result of Boyer, Rolfsen, andWiest [8], which implies that (2 .ρ ) = (2 . LO) for (a superset of) all closed, prime, orientedthree-manifolds. We also have (1) ⇒ (3) for all closed oriented three-manifolds, as shownby Ozsv´ath and Szab´o in the case of C foliations [38], a result recently extended to C foliations by Kazez and Roberts [28], and independently by Bowden [5].More is known for Seifert fibered spaces. For Seifert fibrations over S , we have (1) = (2)as a corollary of a result by Eisenbud, Hirsh, and Neumann [11]. The result that (3) ⇒ (1) isdue to Lisca, Mati´c, and Stipsicz for fibrations over S [31, 32], Boyer, Gordon, and Watsonfor fibrations over RP [7], and Gabai for fibrations with positive first betti number [14].One could also regard the classification by Jankins, Neumann [24], and Naimi [33] of Seifertfibered spaces over S satisfying (1), together with the classification in the present article’sTheorem 5.1 of Seifert fibered L-spaces over S , as an alternative proof that (1) = (3). (cid:3) 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 andClay [6] studied how each of these properties glue together when one splices together Seifertfibered 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 criteriondetermining when they admit compatible gluings. The property (3) of being a non-L-spaceproved 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 mani-folds glued along their torus boundaries have the interiors of their L-space intervals overlapvia the gluing map. In fact, there is no requirement that these Floer simple manifolds begraph manifolds. Theorem 6.2. Suppose that Y and Y are Floer simple manifolds glued together alongtheir boundary tori. Such gluing is specified by a linear map ϕ : H ( ∂Y ) → H ( ∂Y ) with det ϕ = − , descending to a map ϕ P : P ( H ( ∂Y )) → P ( H ( ∂Y )) on Dehn filling slopes.Let I i ⊂ P ( H ( ∂Y i )) denote the interval (with interior ˙ I i ) of L-space filling slopes for Y i ,for each i ∈ { , } , and suppose that ϕ P ( ˙ I ) ∩ ˙ I is nonempty. Then Y ∪ ϕ Y is an L-spaceif and only if ϕ P ( ˙ I ) ∪ ˙ I = P ( H ( ∂Y )) if both D τ ≥ ( Y i ) are nonempty, and if and only if ϕ P ( I ) ∪ I = P ( H ( ∂Y )) otherwise. Set-up for proof: Conventions and simplifying assumptions. We begin bychoosing bases ( m i , l i ) for H ( ∂Y i ) and ¯ m i for H ( Y i ) / Tors( Y i ), for each i ∈ { , } , ac-cording to the conventions of Section 4.1. Thus, if we write ι i : H ( ∂Y i ) → H ( Y i ) for themap induced on homology by inclusion of the boundary, then l i generates ι − ( T i ), where T i := Tors( H ( Y i )), m i satisfies m i · l i = 1, and ¯ m i satisfies ι i ( m i ) ∈ g i ¯ m i + T , where g i := | T ∂i | , with T ∂i := ι ( h l i i ) = T i ∩ ι ( H ( ∂Y i )).We shall break the operation of torus boundary gluing into three steps more amenable toHeegaard Floer computation: those of Dehn filling, connected sum, and Dehn surgery. Inpreparation, assuming ϕ P ( ˙ I ) ∩ ˙ I nonempty, choose µ ∈ P − ( ˙ I ∩ ϕ − P ( ˙ I )) ⊂ H ( ∂Y ) and alongitude λ ∈ H ( ∂Y ) satisfying µ · λ = 1. Set µ := ϕ ( µ ) and λ := − ϕ ( λ ) ∈ H ( ∂Y ),noting that this makes λ a longitude relative to µ , since µ · λ = 1 and det ϕ = − µ · λ = 1. Write µ i = p i m i + q i l i and λ i = q ∗ i m i + p ∗ i l i , with q i q ∗ i − p i p ∗ i = 1, for each i ∈ { , } . Note that the invariant q ∗ := q ∗ p + q ∗ p is independent of choices of µ and λ . That is, if we write ( φ ij ) for the entries of the matrix for ϕ with respect to the bases( m , l ) and ( m , l ), then(90) q ∗ = p q ∗ + q ∗ p = ( φ p + φ q ) q ∗ − ( φ q ∗ + φ p ∗ ) p = − φ . Before using µ i and λ i to splice together Y and Y , we first pause to make some simplifyingassumptions, without loss of generality. Proposition 6.3. Suppose ϕ P ( ˙ I ) ∩ ˙ I = ∅ . For purposes of proving Theorem 6.2, itis sufficient to take q ∗ > , and we may choose µ ∈ P − ( ˙ I ∩ ϕ − P ( ˙ I )) ⊂ H ( ∂Y ) tosatisfy gcd( p i , q i ) = gcd( p , p ) = gcd( p , g ) = gcd( p , g ) = 1 , p , p > q ∗ > , and p i > (1+deg [ ¯ m ] τ c ( Y ))(1+deg [ ¯ m ] τ c ( Y )) for i ∈ { , } , where p i m i + q i l i = µ i , q ∗ i m i + p ∗ i l i = λ i , µ := ϕ ( µ ) , λ := − ϕ ( λ ) , and q ∗ := q ∗ p + q ∗ p for i ∈ { , } . We call such µ “judiciously chosen.” Proof. We summarily dispense with the case in which q ∗ = 0, since then ϕ P ( ˙ I ) ∪ ˙ I = P ( H ( ∂Y )) and Y ∪ ϕ Y is not a rational homology sphere, hence not an L-space. If q ∗ < q ∗ to − q ∗ by making the changes of basis ( m i , l i ) ( m i , − l i ) whilesimultaneously reversing the orientations of both Y and Y . This preserves the positivity of p and p , and leaves invariant the questions of whether Y ∪ ϕ Y is an L-space and whether ϕ P ( ˙ I ) ∩ ˙ I = P ( H ( ∂Y )), or ϕ P ( I ) ∩ I = P ( H ( ∂Y )). Thus we henceforth take q ∗ > µ as an approximation of a primitive rep-resentative P m + Q l ∈ P − ( ˙ I ∩ ϕ − P ( ˙ I )) with P > 0. Since ˙ I ∩ ϕ − P ( ˙ I ) con-tains an open ball, we can demand that P i and Q i are nonzero for i ∈ { , } , where P m + Q l = ϕ ( P m + Q l ). If P < 0, we repair this sign with the change of ba-sis ( m , l ) ( − m , − l ). Writing M ϕ = ( φ ij ) for the matrix for ϕ with respect to thebases ( m , l ) and ( m , l ), choose s ∈ Z such that x := φ + φ s and y := − φ − φ s are nonzero, with gcd( x, g ) = 1, noting that we now have M ϕ ( x, y ) ⊤ = ( − , s ) ⊤ . Next, set(91) D := | g g xy ( yP − xQ )( P + xP ) | , LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 33 and define µ := p m + q l and µ := p m + q l = ϕ ( µ ), with p := P DN + x, p := P DN − , (92) q := Q DN + y, q := Q DN + s for some integer N > q ∗ (1 + deg [ ¯ m ] τ c ( Y ))(1 + deg [ ¯ m ] τ c ( Y )) chosen large enough to make µ := p m + q l lie in P − ( ˙ I ∩ ϕ − P ( ˙ I )). Then gcd( p , g ) = gcd( p , g ) = 1, and one canuse the facts that p /x − q /y = ( yP − xQ )( D/ ( xy )) N is relatively prime to p /x andthat p /x + p = ( P + xP )( D/x ) N is relatively prime to p to argue, respectively, thatgcd( p , q ) = 1 and gcd( p , p ) = 1, the former of which statements implies gcd( p , q ) = 1. (cid:3) Dehn filling a Floer simple manifold. We are now ready to construct Y ∪ ϕ Y asthe Dehn filling of a Floer simple manifold Y . For each i ∈ { , } , perform the (L-space)Dehn filling Y i ( µ i ), writing K µ i for the knot core of Y i ( µ i ) \ Y i . Next, let Y denote the(Floer simple) knot complement(93) Y := Y ( µ ) Y ( µ ) \ K µ K µ of the connected sum K µ K µ ⊂ Y ( µ ) Y ( µ ) = Y ( µ l ), where µ l denotes the meridianof K µ K µ , and as usual, write ι : H ( ∂Y ) → H ( Y ) for the map induced on homology byinclusion of the boundary, and set T := Tors( H ( Y )) and T ∂ := ι ( H ( ∂Y )) ∩ T . The maps f i : H ( Y i ) −→ H ( Y ) induced by inclusion descend to an isomorphism f ⊕ f : ( H ( Y ) ⊕ H ( Y )) / ( ι ( µ ) ∼ ι ( µ )) ∼ −→ H ( Y ) that identifies meridians, via f ι ( µ ) = f ι ( µ ) = ι ( µ l ). In addition, K µ K µ has a longitude λ l satisfying f ( ι ( λ )) + f ( ι ( λ )) = ι ( λ l ).Consider the Dehn filling Y ( λ l ), which one could regard as 0-surgery with respect to thebasis ( µ l , λ l ) along the knot K µ K µ ⊂ Y ( µ l ) = Y ( µ ) Y ( µ ), with Y ( µ l ) an L-space.Since Y already identifies ι ( µ ) with ι ( ϕ ( µ )), and since setting ι ( λ l ) = 0 identifies ι ( λ )with ι ( ϕ ( λ )), we have(94) Y ( λ l ) = Y ∪ ϕ Y . To describe Y ( λ l ) more explicitly, one can deduce that f ⊕ f restricts to an isomorphism(95) ( ι ( H ( ∂Y )) ⊕ ι ( H ( ∂Y ))) / ( ι ( µ ) ∼ ι ( µ )) ∼ −→ ι ( H ( ∂Y )) ⊕ h σ i , for some σ ∈ T with |h σ i| = gcd( g , g ). That is, if we define(96) g := gcd( g , g ) , ˆ g := g /g , ˆ g := g /g , g := g g /g = ˆ g ˆ g g , then for l ∈ H ( ∂Y ) an appropriately signed generator of ι − ( T ) and any m ∈ H ( ∂Y )satisfying m · l = 1, there are σ ∈ T of order g and ξ ∈ Z /g such that f : ι ( m ) p ι ( m ) + q ˆ g ξι ( l ) − q σ , f : ι ( m ) p ι ( m ) + q ˆ g ξι ( l ) + q σ , (97) f : ι ( l ) p ˆ g ξι ( l ) + p σ , f : ι ( l ) p ˆ g ξι ( l ) − p σ . Thus, g = | T ∂ | , and if we write(98) µ l = pm + ql, λ l = q ∗ m + p ∗ l, then p , q , q ∗ , and p ∗ satisfy p = p p , q ≡ ( q p g + q p g ) ξ (mod g ) , (99) q ∗ = q ∗ p + q ∗ p , p ∗ ≡ (( p p ∗ + q ∗ q ) g + ( p p ∗ + q ∗ q ) g ) ξ (mod g ) . Again, the condition µ l · λ l = 1 determines the value of ξ , which we shall not need. Ofcourse, it will often be more convenient to express this restriction of ι i ( H ( ∂Y i )) to f ⊕ f 24 JACOB RASMUSSEN AND SARAH DEAN RASMUSSEN in terms of the bases ( ι i ( µ i ) , ι i ( λ i )) for ι i ( H ( ∂Y i )) and ( ι ( µ l ) , ι ( λ l )) for ι ( H ( ∂Y )), as weshall describe explicitly in the proof of Proposition 6.5.In either case, we see that q ∗ = q ∗ p + q ∗ p makes its appearance as λ l · l . Thus, Y ∪ ϕ Y = Y ( λ l ) can be regarded as surgery with label ( µ l · λ l ) / ( λ l · l ) = 1 /q ∗ along K µ K µ ⊂ Y ( µ l ).6.5. Computation of D τ ( Y ) . For the remainder of Section 6, we regard the entire pre-ceding construction, along with the hypotheses of Theorem 6.2, as fixed initial data. Weare now ready to compute D τ ( Y ), which we shall call D τ ≥ ( Y ) to emphasize that in this casewe are not excluding torsion elements. Proposition 6.4. Suppose that µ is “judiciously chosen” from P − ( ˙ I ∩ ϕ − P ( ˙ I )) nonempty,and that Y is constructed as above. If we set t ∂ := [( ι ( m ))] , then D τ ≥ ( Y ) = A ∐ ( A ∪ A ) ∐ A , with A := S (cid:20) − t ∂ − − t p p ∂ (1 − t p ∂ )(1 − t p ∂ ) (cid:21) + T ∂ , (100) A := f ( D τ ≥ ( Y )) + f (cid:0) { , . . . , p − } ι ( m ) + T ∂ (cid:1) ,A := f ( D τ ≥ ( Y )) + f (cid:0) { , . . . , p − } ι ( m ) + T ∂ (cid:1) ,A := ι ( µ l ) + f ( D τ ≥ ( Y )) + f ( D τ ≥ ( Y )) . Proof. To compute D τ ≥ ( Y ), we need the Turaev torsion τ ( Y ) and torsion complement τ c ( Y ). In order to write these down, we first choose generators ¯ m for H ( Y ) /T and ¯ m i and H ( Y i ) /T i satisfying(101) ι ( m ) ∈ g ¯ m + T, ι i ( m i ) ∈ g i ¯ m i + T i , i ∈ { , } . Recall that the above condition only constrains the signs of ¯ m and ¯ m i . We shall write(102) t := [ ¯ m ] ∈ Z [ H ( Y )] , t i := [ ¯ m i ] ∈ Z [ H ( Y i )] , i ∈ { , } , for the inclusions of ¯ m and ¯ m i into their respective group rings.Invoking the standard gluing rules for Turaev torsion yields τ ( Y ) = (1 − [ ι ( µ l )]) ˜ f ( τ ( Y )) ˜ f ( τ ( Y )) , (103)where each ˜ f i denotes the lift of f i to the Laurent series group ring Z [ t − i , t i ]][ T i ] ⊃ Z [ H ( Y i )].(One could also obtain this result by using Proposition 2.1 and the fact that Heegaard Floerhomology tensors on connected sums.)For i ∈ { , } , set P T := P h ∈ T [ h ] ∈ Z [ H ( Y )] and P T i := P h i ∈ T i [ h i ] ∈ Z [ H ( Y i )], andlet P and P i denote the Laurent series P := P T / (1 − t ) and P i := P T i / (1 − t i ), the latterwith polynomial truncations(104) ¯ P i := (1 − [ ι i ( µ i )]) P i = 1 − t p i g i i − t i P T i . The torsion complements τ c ( Y ) := P − τ ( Y ) and τ c ( Y i ) := P i − τ ( Y i ) then satisfy τ c ( Y ) = P − (1 − [ ι ( µ l )]) ˜ f ( P − τ c ( Y )) ˜ f ( P − τ c ( Y ))(105) = A c0 + A c12 + A c3 , with A c0 := P − (1 − [ ι ( µ l )]) ˜ f ( P ) ˜ f ( P ) ,A c12 := ˜ f ( τ c ( Y )) ˜ f ( ¯ P ) + ˜ f ( ¯ P ) ˜ f ( τ c ( Y )) − ˜ f ( τ c ( Y )) ˜ f ( τ c ( Y )) ,A c3 := [ ι ( µ l )] ˜ f ( τ c ( Y )) ˜ f ( τ c ( Y )) . LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 35 It is straightforward to show that each of A c0 , A c12 , and A c3 is an element of Z [ H ( Y )]with coefficients in { , } , and that the three sets S [ A c0 ], S [ A c12 ], and S [ A c3 ] are disjoint. Inparticular, A c0 satisfies the property(106) (cid:16) S [ A c0 ] − S [ ˜ f ( P ) ˜ f ( P )] (cid:17) ∩ S [ P ] = S [ A c0 ] , while A c12 satisfies(107) S [ A c12 ] = S [ ˜ f ( τ c ( Y )) ˜ f ( ¯ P ) + ˜ f ( ¯ P ) ˜ f ( 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(108) S [ τ ( Y )] = S [ ˜ f ( τ ( Y )) ˜ f ( τ ( Y ))] ⊂ H ( Y ) . Lastly, we compute D τ ≥ ( Y ) := ( S [ τ c ( Y )] − S [ τ ( Y )]) ∩ ι ( m Z ≥ + l Z ). Using the facts that0 ∈ S [ τ ( Y i )] for each i ∈ { , } (as per the convention stated in (7) in Section 4.2) and that ι ( H ( ∂Y )) ⊂ f ι ( H ( ∂Y )) ⊕ f ι ( H ( ∂Y )), we obtain D τ ≥ ( Y ) = A ∐ ( A ∪ A ) ∐ A , with A = S [ A c0 ] ∩ ι ( m Z ≥ + lZ ) , (109) A = f ( D τ ≥ ( Y )) + f ( S [ ¯ P ] ∩ ι ( H ( Y ))) ,A = f ( D τ ≥ ( Y )) + f ( S [ ¯ P ] ∩ ι ( H ( Y ))) ,A = ι ( µ l ) + f ( D τ ≥ ( Y )) + f ( D τ ≥ ( Y )) , where property (106) has made any remaining subsets of S [ τ c ( Y )] − S [ τ ( Y )]—such as, forexample, f ( S [ τ c ( Y )] − S [ τ ( Y )]) ∩ ( m Z < + T )—land in S [ A c0 ]. It is straightforward to showthat the above A i are equal to those enumerated in the statement of the proposition. (cid:3) Computation of L-space interval for Y . Having determined D τ ( Y ), we can applyTheorem 4.2 to compute the L-space interval for Y . Proposition 6.5. Suppose that µ is “judiciously chosen” from P − ( ˙ I ∩ ϕ − P ( ˙ I )) nonempty,and that Y is constructed as above. For each i ∈ { , } , set ¯ q i := [ q ∗ i ] p i and let B i denote theset B i := (cid:8) [ p i γ i − q i δ i ] p i g i | δ i = δ i ι i ( m i ) + γ i ι i ( l i ) ∈ D τ ≥ ( Y i ) (cid:9) . Then Y ∪ ϕ Y is an L-spaceif and only if condition ( l .i ) holds for each b ∈ B , ( l .ii ) holds for each b ∈ B , and ( l .iii ) holds for each ( b , b ) ∈ B × B with b ≡ b (mod g ) : ( l .i ) 1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) ≥ ∀ b ≡ b (mod p g ) , < b < pg, ( l .ii ) 1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) ≥ ∀ b ≡ b (mod p g ) , < b < pg, ( l .iii ) 1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) > ∀ b ≡ b (mod p g ) , b ≡ b (mod p g ) , < b < pg, where p := p p and g := g g /g , with g = gcd( g , g ) .Proof. We begin by ensuring that D τ ≥ ( Y ) meets the conditions of Theorem 4.2 Since A T implies D τ > ( Y ) = ∅ , it remains to verify, for each δ = δι ( m ) + γι ( l ) ∈ D τ ≥ ( Y ), that b δ := [ pγ − qδ ] pg ( ≡ µ l · ι − ( δ ) (mod pg )) is nonzero, or equivalently, that δ / ∈ h ι ( µ l ) i . Now,the definition of D τ ≥ already implies 0 / ∈ D τ ≥ ( Y ). Recalling the result of Proposition 6.4, and that ι ( µ l ) = pι ( m ) + qι ( l ) with p := p p , we know that the inclusions A ⊂ { , . . . , p p − p − p } ι ( m ) + T ∂ , (110) A ∪ A ⊂ f (cid:0) { , . . . , p − } ι ( m ) + T ∂ (cid:1) + f (cid:0) { , . . . , p − } ι ( m ) + T ∂ (cid:1) (111) = ( { , . . . , p − } p + { , . . . p − } p ) ι ( m ) + T ∂ imply that h ι ( µ l ) i ∩ ( A ∪ A ∪ A ) = ∅ . Lastly, since our “judiciously chosen” hypothesismakes deg [ ¯ m i ] τ c ( Y i ) < p i g i = deg [ ¯ m i ] [ ι i ( µ i )], and since the kernel of f ⊕ f is generated by( ι ( µ ) , − ι ( µ )), we know that h ι ( µ l ) i ∩ A = ∅ . Thus, Theorem 4.2 applies.Since we can regard Y ∪ ϕ Y = Y ( λ l ) as surgery with label 1 /q ∗ along K µ K µ ⊂ Y ( µ l ),Theorem 4.2 tells us that Y ∪ ϕ Y is an L-space if and only if(112) b δ − pδ ≤ q ∗ ≤ b δ δ for all δ = δι ( m ) + γι ( l ) ∈ D τ > ( Y ) (= D τ ≥ ( Y ) \ T ). Now, since b δ ≡ µ l · ˜ δ (mod pg ) for anylift ˜ δ ∈ ι − ( 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 < q ∗ > 0, the left-hand inequality in (112) is vacuous, whereas the right-handinequality is equivalent to the condition a δ ≤ b δ q ∗ > δ = δι ( m ) + γι ( l ) ∈ D τ ≥ ( Y ), we obtain a δ ≤ δ < p . In particular, a δ ≤ δ ∈ A and for any δ ∈ D τ ≥ ( Y ) ∩ (0 ι ( m ) + T ∂ ).Now, the latter case is, strictly speaking, irrelevant to the question of whether Y ∪ ϕ Y isan L-space, but the fact that the condition a δ ≤ D τ ≥ ( Y )allows us to apply the condition to all of D τ ≥ ( Y ), thereby simplifying our bookkeeping.It remains to apply the condition a δ ≤ A , A , and A , from which weshall obtain the respective conditions ( l .i ), ( l .ii ), and ( l .iii ). To do this, we first, for each i ∈ { , } , consider the bijection, { , . . . , p i g i − } −→ { , . . . , p i − } ι i ( m i ) + T ∂i ⊂ ι i ( H ( ∂Y i )) , (113) b i ι i ( − (cid:22) b i q ∗ i p i (cid:23) µ i + b i λ i ) ∈ [ b i ¯ q i ] p i ι i ( m i ) + T ∂i , recalling that µ i = p i m i + q i l i , λ i = q ∗ i m i + p ∗ i l i , and g i := | T ∂i | with T ∂i = h ι i ( l i ) i . Theinverse map sends(114) x i := x i ι i ( m i ) + y i ι i ( l i ) b x i i := [ µ i · ( x i m i + y i l i )] p i g i = [ p i y i − q i x i ] p i g i . Thus, if we define a x i i := − ( b x i i q ∗ i − [ b x i i q ∗ i ] p i ) /p i , then for any x i := x i ι i ( m i ) + y i ι i ( l i ) with x i ∈ { , . . . , p i − } , and for any s i ∈ Z , we have(115) x i = ι i ( a x i i µ i + b x i i λ i ) = ι i (( a x i i − q ∗ i g i s i ) µ i + ( b x i i + p i g i s i ) λ i ) , with s i ∈ Z parametrizing the lifts ι − i ( x i ) of x i .Since f ι ( µ ) = f ι ( µ ) = ι ( µ l ) and f ι ( λ ) + f ι ( λ ) = ι ( λ l ), we deduce from(115) that f ( x ) + f ( x ) ∈ ι ( H ( ∂Y )) if and only if there exist s , s ∈ Z such that b x + p g s = b x + p g s , which, in turn, occurs if and only if b x ≡ b x (mod g ), since g = gcd( p g , p g ) = gcd( g , g ). In such case, if we write f ( x ) + f ( x ) = ι ( aµ l + bλ l )with b ∈ { , . . . , pg − } , then b is the unique solution in { , . . . , pg − } to the equivalences b ≡ b x (mod p g ), b ≡ b x (mod p g ). Setting b = b x i i + p i g i s i makes g i s i = ( b − b x i i ) /p i LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 37 for each i ∈ { , } , so that we obtain a = P i ∈{ , } ( a x i i − q ∗ i g i s i )(116) = P i ∈{ , } (cid:16) − ( b x i i q ∗ i − [ b x i i q ∗ i ] p i ) /p i − q ∗ i ( b − b x i i ) /p i (cid:17) = − b (¯ q p + ¯ q p − p p ) /p p + [ b x q ∗ ] p /p + [ b x q ∗ ] p /p = − b − (cid:22) b ¯ q p (cid:23) − (cid:22) b ¯ q p (cid:23) , where the third line uses the identity q ∗ := q ∗ p + q ∗ p = ¯ q p + ¯ q p − p p , implied by thehypothesis p p > q ∗ > q i := [ q ∗ i ] p i .Since we may write any δ ∈ A as(117) δ = f ( δ ) + f ( x ) = ι ( aµ l + bλ l )with δ ∈ D τ ≥ ( Y ), x ∈ { , . . . , p − } ι ( m ) + { , . . . , g − } ι ( l ) satisfying b x ≡ b δ (mod g ), 0 < b < pg , and a as determined in (116), we have a δ = a , and demanding a δ ≤ i ) l . Likewise, applying a δ ≤ δ ∈ A yields condition( ii ) l . The case of A is similar, except that since δ = ι ( µ l ) + f ( δ ) + f ( δ ), we need a δ = 1 + a ≤ b δ ≡ b δ (mod g ), yielding condition (iii l ). (cid:3) Determining when gluing hypothesis is met. We next turn our attention to theL-space filling slope intervals I i ⊂ P ( H ( ∂Y i )), to determine when they combine accordingto the hypotheses of the theorem. Proposition 6.6. Suppose that µ is “judiciously chosen” from P − ( ˙ I ∩ ϕ − P ( ˙ I )) nonempty,and that Y is constructed as above. For each i ∈ { , } , set ¯ q i := [ q ∗ i ] p i and let B i de-note the set B i := (cid:8) [ p i γ i − q i δ i ] p i g i | δ i = δ i ι i ( m i ) + γ i ι i ( l i ) ∈ D τ ≥ ( Y i ) (cid:9) . Then ϕ P ( ˙ I ) ∪ ˙ I = P ( H ( ∂Y )) when both D τ ≥ ( Y i ) are nonempty—and ϕ P ( I ) ∪ I = P ( H ( ∂Y )) when one orboth D τ ≥ ( Y i ) are empty—if and only if the following three conditions hold: ( i .i ) 1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) ≥ b ∈ B , ( i .ii ) 1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) ≥ b ∈ B , ( i .iii ) 1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) > b , b ) ∈ B × B . Proof. For i ∈ { , } , let π i denote the “surgery label” map, π i : H ( ∂Y i ) \ { } −→ Q ∪ ∞ ,(118) π i : α i µ i + β i λ i µ i · ( α i µ i + β i λ i )( α i µ i + β i λ i ) · l i = β i α i p i + β i q ∗ i , and for each δ i = δ i ι i ( m i ) + γ i ι i ( l i ) ∈ D τ ≥ ( Y i ), let ˜ δ i + , ˜ δ i − ∈ ι − i ( δ i ) denote the two lifts of δ i closest to µ i with respect to surgery label, i.e. ,˜ δ i + = a δ i i + µ i + b δ i i + λ i , b δ i i + := [ p i γ i − q i δ i ] p i g i (119) ˜ δ i − = a δ i i − µ i + b δ i i − λ i := (cid:16) a δ i i + + q ∗ i g i (cid:17) µ i + (cid:16) b δ i i + − p i g i (cid:17) λ i . (120)Note that since p i > deg [ ¯ m i ] τ c ( Y i ) implies δ i < p i , we have(121) δ i = a δ i i + p i + b δ i i + q ∗ i = [ b δ i i + q ∗ i ] p i = a δ i i − p i + b δ i i − q ∗ i = [ b δ i i − q ∗ i ] p i ≥ . Note also that π i (˜ δ i − ) < < π i (˜ δ i + ) unless δ i = 0, in which case π i (˜ δ i − ) = π i (˜ δ i + ) = ∞ .Corollary 4.4 then implies that, for D τ ≥ ( Y i ) nonempty, ˜ I i := P − ( I i ) ⊂ H ( ∂Y i ) takes theform ˜ I i = T δ i ∈D τ ≥ ( Y i ) ˜ I δ i i , where(122) ˜ I δ i i := (cid:26) µ ∈ H ( ∂Y i ) \ { } (cid:12)(cid:12)(cid:12)(cid:12) π i (˜ δ i − ) ≤ π i ( µ ) ≤ π i (˜ δ i + ) if δ i > ,π i ( µ ) = ∞ (= π i (˜ δ i − ) = π i (˜ δ i + )) if δ i = 0 (cid:27) . If D τ ≥ ( Y i ) = ∅ , then, similarly to the case in which δ i = 0, ˜ I i is the complement of π − i ( ∞ ).Note that we always have ∞ / ∈ π i ( ˜ I i ). Thus, a necessary condition to achieve ϕ P ( I ) ∪ I = P ( H ( ∂Y )) or ϕ P ( ˙ I ) ∪ ˙ I = P ( H ( ∂Y )) is to have(123) ( ∞ .i ) ∞ ∈ π ◦ ϕ ( ˜ I ) , ( ∞ .ii ) ∞ ∈ π ◦ ϕ − ( ˜ I ) . We claim that conditions ( ∞ .i ) and ( ∞ .ii ) are respectively equivalent to ( i .i ) and ( i .ii ). Firstnote that it is sufficient to prove the equivalence of ( ∞ .i ) and ( i .i ), since the maps(124) ϕ : αµ + βλ αµ − βλ , ϕ − : αµ + βλ αµ − βλ are exchanged by swapping i = 1 with i = 2. Also, when D τ ≥ ( Y ) = ∅ , in which case ( i .i )holds vacuously, our hypothesis that q ∗ = 0, ensuring that π ϕπ − ( ∞ ) = ∞ , implies ( ∞ .i )holds automatically. Thus, we henceforth assume that D τ ≥ ( Y ) is nonempty.For any a µ + b λ ∈ H ( ∂Y ) \ { } , it is straightforward to show that the map(125) π ◦ ϕ : a µ + b λ 7→ − b a p − b q ∗ has denominator satisfying(126) a p − b q ∗ = p p ( a p + b q ∗ ) − b q ∗ p . In particular, since q ∗ > 0, and since δ = a δ − p + b δ − q ∗ ≥ b δ − < δ ∈ D τ ≥ ( Y ), we have(127) a δ − p − b δ − q ∗ > , π ◦ ϕ (˜ δ − ) > δ ∈ D τ ≥ ( Y ) . Now, there are two ways in which π ◦ ϕ ( ˜ I δ ) could contain ∞ . One is if ∞ is contained asan endpoint of π ◦ ϕ ( ˜ I δ ), in which case, since π ◦ ϕ (˜ δ − ) = ∞ , we must have π ◦ ϕ (˜ δ ) = ∞ , or equivalently, a δ p − b δ q ∗ = 0. Conveniently, the condition π ◦ ϕ (˜ δ − ) = π ◦ ϕ (˜ δ )also implies that π ◦ ϕ ( ˜ I δ ) is closed in this case. The other possibility is that ∞ lies in theinterior of π ◦ ϕ ( ˜ I δ ). Since π − , ϕ , and π are each orientation reversing, this is equivalentto the condition that π ◦ ϕ (˜ δ − ) ≤ π ◦ ϕ (˜ δ ), which, since π ◦ ϕ (˜ δ − ) > 0, implies π ◦ ϕ (˜ δ ) > a δ p − b δ q ∗ < 0. In fact, the converse is also true: using thesubstitutions a δ − = a δ + q ∗ g and b δ − = b δ − p g , and the fact that a δ p + b δ q ∗ ≥ a δ − p − b δ − q ∗ > a δ p − b δ q ∗ < π ◦ ϕ (˜ δ − ) ≤ π ◦ ϕ (˜ δ ), Thus, in summary, ( ∞ .i ) holds ifand only if a δ p − b δ q ∗ ≤ δ ∈ D τ ≥ ( Y ), or equivalently, if and only if(128) a δ p − b δ q ∗ ≤ − [ b δ q ∗ ] p for all δ ∈ D τ ≥ ( Y ) , which, after substituting a δ = ([ b δ q ∗ ] p − b δ q ∗ ) /p and q ∗ p + q ∗ p = ¯ q p + ¯ q p − p p ,becomes condition ( i .i ).Thus, conditions ( ∞ .i ) and ( ∞ .ii ) are respectively equivalent to conditions ( i .i ) and ( i .ii ).When one or both of D τ ≥ ( Y i ) are empty, ( i .iii ) holds vacuously, and ( ∞ .i ) and ( ∞ .ii ) are LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 39 jointly equivalent to the condition that ϕ P ( I ) ∪ I = P ( H ( ∂Y )). We henceforth assumethat each D τ ≥ ( Y i ) = ∅ , and that conditions ( i .i ) and ( i .ii ), hence ( ∞ .i ) and ( ∞ .ii ), hold.For each ( δ , δ ) ∈ D τ ≥ ( Y ) × D τ ≥ ( Y ), the substitutions a δ = ([ b δ q ∗ ] p − b δ q ∗ ) /p , a δ = ([ b δ q ∗ ] p − b δ q ∗ ) /p , and q ∗ p + q ∗ p = ¯ q p + ¯ q p − p p make the condition(129) 1 b δ $ b δ ¯ q p % + 1 b δ $ b δ ¯ q p % > a δ b δ + a δ b δ < , which, after we multiply by − p and add b δ ( a δ p − b δ q ∗ ) to both sides, becomes(131) − b δ ( a δ p + b δ q ∗ ) > b δ ( a δ p − b δ q ∗ ) , which, since − b δ ( a δ p + b δ q ∗ ) ≤ 0, implies a δ p − b δ q ∗ = 0, and hence π ◦ ϕ (˜ δ ) = ∞ .Note that when π ◦ ϕ (˜ δ ) = ∞ , condition ( ∞ .i ) is equivalent to the condition(132) (0 < ) π ◦ ϕ (˜ δ − ) ≤ π ◦ ϕ (˜ δ ) . If δ = 0, then ˜ I δ is the complement of π − ( ∞ ), and so (132) is equivalent to thecondition that ϕ P ( ˙ I δ ) ∪ ˙ I δ = P ( H ( ∂Y )), where ˙ I δ i i denotes the interior of P ( ˜ I δ i i ) for each i ∈ { , } . If δ > 0, so that π (˜ δ − ) < < π (˜ δ ), then dividing (131) by δ ( a δ p − b δ q ∗ )makes (131) equivalent to the inequality π ◦ ϕ (˜ δ ) < π (˜ δ ), which, combined with (132),becomes(133) π (˜ δ − ) < < π ◦ ϕ (˜ δ − ) ≤ π ◦ ϕ (˜ δ ) < π (˜ δ ) , which again is equivalent to the condition that ϕ P ( ˙ I δ ) ∪ ˙ I δ = P ( H ( ∂Y )). Thus condition( i .iii ), which takes (129) over all ( δ , δ ) ∈ D τ ≥ ( Y ) × D τ ≥ ( Y ), is equivalent to the conditionthat ϕ P ( ˙ I ) ∪ ˙ I = P ( H ( ∂Y )). (cid:3) Comparison of L-space classification with gluing hypothesis. Now that wehave both classified when Y ∪ ϕ Y is an L-space, and classified when it satisfies the gluinghypothesis in terms of the union of the L-space intervals of Y and Y , it remains to showthat these two classifications are equivalent. Proposition 6.7. Suppose that µ is “judiciously chosen” from P − ( ˙ I ∩ ϕ − P ( ˙ I )) nonempty,and that Y is constructed as above. For each i ∈ { , } , set ¯ q i := [ q ∗ i ] p i and let B i denotethe set B i := (cid:8) [ p i γ i − q i δ i ] p i g i | δ i = δ i ι i ( m i ) + γ i ι i ( l i ) ∈ D τ ≥ ( Y i ) (cid:9) . Then condition ( i .i ) (re-spectively ( 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 ∈ B (respectively b ∈ B ).Proof. If B = ∅ , then conditions ( i .i ) and ( i .ii ) hold vacuously, hence are equivalent. Wetherefore assume B is nonempty and fix some b ∈ B . Clearly ( l .i ) implies the statementof ( i .i ) for that particular b , since b ∈ { b ∈ Z | b ≡ b (mod p ) , < b < p g p g /g } .Conversely, suppose ( i .i ) holds for that b . Substituting q ∗ = ¯ q p + ¯ q p − p p gives(134) b q ∗ p p ≥ [ b ¯ q ] p p + [ b ¯ q ] p p . Thus, for any b := b + yp g with y ∈ { , . . . , p g /g − } , we have bq ∗ p p ≥ (cid:18) [ b ¯ q ] p p + [ b ¯ q ] p p (cid:19) + yp g ( p ¯ q − ( p − ¯ q ) p ) p p (135) ≥ [ b ¯ q ] p p + (cid:18) [ b ¯ q ] p p + yg [ p ¯ q ] p p (cid:19) ≥ [ b ¯ q ] p p + [ b ¯ q ] p p , which is equivalent to the inequality in condition ( l .i ). An analogous argument proves theequivalence of conditions ( l .ii ) and ( i .ii ) for any b ∈ B . (cid:3) Proposition 6.8. Suppose that µ is “judiciously chosen” from P − ( ˙ I ∩ ϕ − P ( ˙ I )) nonempty,and that Y is constructed as above. For each i ∈ { , } , set ¯ q i := [ q ∗ i ] p i and let B i denotethe set B i := (cid:8) [ p i γ i − q i δ i ] p i g i | δ i = δ i ι i ( m i ) + γ i ι i ( l i ) ∈ D τ ≥ ( Y i ) (cid:9) . Suppose conditions ( i .i ) and ( i .ii ) from Proposition 6.6 hold. Then condition ( i .iii ) from Proposition 6.6 holds ifand only if condition ( l .iii ) from Proposition 6.5 holds for all ( b , b ) ∈ B × B with b ≡ b (mod g ) .Proof. We henceforth assume that D τ ≥ ( Y ) and D τ ≥ ( Y ) are nonempty, since otherwise con-ditions ( i .iii ) and ( l .iii ) hold vacuously in all cases.If condition ( i .iii ) holds, then it holds for any ( b , b ) ∈ B × B with b ≡ b (mod g ).In this case, the unique b ∈ { , . . . , p p g − } satisfying b ≡ b (mod p g ) and b ≡ b (mod p g ) also satisfies [ b ¯ q ] p = [ b ¯ q ] p and [ b ¯ q ] p = [ b ¯ q ] p , so that we have1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) = 1 b (cid:22) b ¯ q p (cid:23) + 1 b (cid:22) b ¯ q p (cid:23) + (cid:18) b − b (cid:19) [ b ¯ q ] p p + (cid:18) b − b (cid:19) [ b ¯ q ] p p (136) > . Thus ( i .iii ) implies ( l .iii ) for all ( b , b ) ∈ B × B with b ≡ b (mod g ), and it remainsto prove the converse. Claim. Suppose that conditions ( i .i ) and ( i .ii ) hold, and that there exists some ( b , b ) ∈ B × B for which the statement of ( i .iii ) fails, or equivalently (using the substitution q ∗ =¯ q p + ¯ q p − p p , for which (137) q ∗ p p ≤ [ b ¯ q ] p b p + [ b ¯ q ] p b p . Then we have the inequalities (138) ( i ) [ b ¯ q ] p b ≥ [ b ¯ q ] p b , ( ii ) [ b ¯ q ] p b ≥ [ b ¯ q ] p b , and conditions ( i .i ) and ( i .ii ) for this particular ( b , b ) ∈ B × B become the equalities (139) ( i ) q ∗ p p = [ b ¯ q ] p b p + [ b ¯ q ] p b p , ( ii ) q ∗ p p = [ b ¯ q ] p b p + [ b ¯ q ] p b p . Proof of Claim. Using the substitution q ∗ = ¯ q p + ¯ q p − p p , we can re-expressconditions ( i .i ) and ( i .ii ) as(140) ( i ) q ∗ p p ≥ [ b ¯ q ] p b p + [ b ¯ q ] p b p , ( ii ) q ∗ p p ≥ [ b ¯ q ] p b p + [ b ¯ q ] p b p . LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 41 Concatenating (137) with (140 .i ) (respectively, (140 .ii )) then yields inequality (138 .ii ) (re-spectively, (138 .i )). Setting δ := [ b ¯ q ] p ∈ D τ ≥ ( Y ) and δ := [ b ¯ q ] p ∈ D τ ≥ ( Y ), we notethat (138 .i ) implies(141) δ b p < b , since otherwise, applying (138 .i ) and 1 /b ≤ δ / ( b p ) in succession would yield[ b ¯ q ] p b p ≤ δ p · b ≤ δ p · δ b p = δ δ /p b p < (1 + deg t τ c ( Y ))(1 + deg t τ c ( Y ) /p b p < b p , making [ b ¯ q ] p < 1, a contradiction. Thus (141) must hold.Applying (140 .i ), (137), and (141) in succession, we obtain[ b ¯ q ] p b p + [ b ¯ q ] p b p ≤ q ∗ p p (142) ≤ [ b ¯ q ] p b p + [ b ¯ q ] p b p < [ b ¯ q ] p b p + 1 b . (143)Subtracting [ b ¯ q ] p b p + [ b ¯ q ] p b p from lines (142) and (143) then yields(144) 0 ≤ q ∗ p p − (cid:18) [ b ¯ q ] p b p + [ b ¯ q ] p b p (cid:19) < b − [ b ¯ q ] p b p , but we also know that(145) q ∗ p p − (cid:18) [ b ¯ q ] p b p + [ b ¯ q ] p b p (cid:19) ∈ b Z . Thus, (139 .i ) must hold, and (139 .ii ) follows from symmetry, proving our Claim.Having proven our Claim, we pause to introduce the notation b i δ b i i for the bijection { , . . . , p i g i − } → { , . . . , p i − } ι i ( m i ) + T ∂i , (146) b i δ b i i := ι i ( − (cid:22) b i q ∗ i p i (cid:23) µ i + b i λ i ) ∈ [ b i ¯ q i ] p i ι i ( m i ) + T ∂i , whose inverse we used to define each B i as a set of integers indexing the elements of D τ ≥ ( Y i ).We now proceed with an inductive argument. Suppose that ( l .iii ) holds for all ( b , b ) ∈ B × B satisfying b ≡ b (mod g ), and that ( i .i ) and ( i .ii ) hold, but that there exist b i ∈ B i and b I ∈ B I , with { i, I } = { , } and b i ≤ b I , for which ( i .iii ) fails, i.e. , for which(147) q ∗ p p ≤ [ b i ¯ q i ] p i b i p i + [ b I ¯ q I ] p I b I p I . Equation (139) from our Claim then tells us that(148) 1 b i (cid:22) b i ¯ q i p i (cid:23) + 1 b i (cid:22) b i ¯ q I p I (cid:23) = 1 . This means that b i / ∈ B I , since otherwise, setting b := b i ∈ B i ∩ B I = B ∩ B would make(148) contradict condition ( l .iii ). Thus, δ b i I / ∈ D τ ≥ ( Y I ) and b i < b I . We next apply (138) from our Claim, to obtain(149) [ b i ¯ q I ] p I ≤ b i b I [ b I ¯ q I ] p I < [ b I ¯ q I ] p I . Since δ b I I − δ b i I ∈ ([ b I ¯ q I ] p I − [ b i ¯ q I ] p I ) ι I ( m I ) + T ∂I , the above inequality implies δ b I I − δ b i I ∈ ι I ( m I Z ≥ + l I Z ). Thus, since δ b i I / ∈ D τ ≥ ( Y I ) and δ b I I ∈ D τ ≥ ( Y I ), the additive closure of ι I ( m I Z ≥ + l I Z ) \ D τ ≥ ( Y I ) from Proposition 4.1 tells us that δ b I I − δ b i I ∈ D τ ≥ ( Y I ). Since (149)implies ([ b I ¯ q I ] p I − [ b i ¯ q I ] p I ) = [( b I − b i )¯ q I ] p I , we actually have δ b I I − δ b i I = δ b I − b i I ∈ D τ ≥ ( Y I ),implying b I − b i ∈ B i . We furthermore have(150) [ b i ¯ q I ] p I b i ≤ [ b I ¯ q I ] p I b I = ⇒ [ b I ¯ q I ] p I b I ≤ [( b I − b i )¯ q I ] p I b I − b i , so that (147) implies(151) q ∗ p p ≤ [ b i ¯ q i ] p i b i p i + [( b I − b i )¯ q I ] p I ( b I − b i ) p I , with b i ∈ B i and b I − b i ∈ B I , mimicking our initial conditions.We then iterate the process, at each iteration redefining i, I ∈ { , } , b i , and b I so that(152) b new i := min { b old i , b old I − b old i } , b new I := max { b old i , b old I − b old i } . Like any Euclidean Algorithm, this strictly decreasing sequence bounded by zero mustterminate at zero, with its last two nonzero entries equal to(153) b final i = b final I = gcd( b original i , b original I ) . Setting b := b final i = b final I ∈ B ∩ B then makes (148) contradict condition ( l .iii ),This completes the proof of the proposition, thereby completing the proof of Theorem 6.2 (cid:3) Generalized solid tori and NLS Detection In this section, we study manifolds with D τ > = ∅ . Unless otherwise specified, we assumethat Y is a rational homology S × D with H ( Y ) = Z ⊕ T , and that φ : H ( Y ) → H ( Y ) /T ≃ Z is the projection. We define g Y > φ = g Y Z ⊂ Z . Thenumber g Y is the minimal intersection number of a curve on ∂Y with a surface generating H ( Y, ∂Y ). Equivalently, it is the minimal number of boundary components of such asurface, or the order of the homological longitude l in H ( Y ). Finally, we define k Y to bethe order of the group T / ( T ∩ im ι ), so that | T | = k Y g Y .7.1. Generalized solid tori. The Seifert fibred spaces N g = M ( ∅ ; 1 /g, − /g ) provide amotivating example of a class of manifolds with L ( N g ) = Sl ( N g ) / [ l ]. They were studied in[7] (for g = 2) and subsequently by Watson [18] for arbitrary values of g . We briefly describethem here. First, we have H ( N g ) = h f, h , h | f + gh = f − gh = 0 i ≃ Z ⊕ Z /g. The Z summand is generated by h , and the Z /g summand is generated by σ = h + h . H ( ∂N g ) = h f, σ i , so ι ( H ( ∂N g )) = g Z ⊕ Z /g ⊂ H ( Y g ). The Turaev torsion is τ ( N g ) ∼ − [ f ](1 − [ h ])(1 − [ h ]) = 1 − t g (1 − t )(1 − tσ ) LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 43 so the Milnor torsion is τ ( Y ) = τ ( N g ) | σ =1 = 1 − t g (1 − t ) = 1 + 2 t + 3 t + . . . + ( g − t g − + gt g + gt g +1 + . . . It is easy to see that if x S [ τ ( N g )], y ∈ S [ τ ( N g )] with φ ( x ) > φ ( y ), then φ ( x − y ) < g .If x − y ∈ im ι , we must have φ ( x − y ) = 0, so D τ > ( N g ) = ∅ . More generally, the sameargument shows that Proposition 7.1. If Y is a Floer simple and deg ∆( Y ) < g Y , then D τ > ( Y ) = ∅ . Motivated by this, we make the following Definition 7.2. A generalized solid torus is a Floer simple manifold Y with deg ∆( Y ) < g Y . If Y is such a manifold, Corollary 2.3 implies that k Y k ≤ g Y − 2. On the other hand,an embedded surface which generates H ( Y, ∂Y ) has at least g Y boundary components, soa norm-minimizing surface must have genus 0.The Milnor torsion of a generalized solid torus is determined by g Y and k Y . Lemma 7.3. Suppose that Y is a rational homology S × D . If p : Z [ t ] → Z [ t ] / ( t g Y − is the projection, then p (∆( Y )) = k Y (1 − t g Y ) / (1 − t ) . Proof. The usual product formula for the torsion implies that τ ( Y ( l )) = j ∗ ( τ ( S × D ) j ∗ ( τ ( Y ))where j : S × D → Y ( l ) and j : Y → Y ( l ) are the inclusions. It follows that τ ( Y ( l )) = τ ( Y )1 − t g Y . By [45], Lemma 3.2, we have τ ( Y ( l )) = t c | H ( Y ( l ) | (1 − t ) + P ( t )where c ∈ Z and p ( t ) ∈ Z [ t ± ]. | H ( Y ( l )) | = k Y . Combining the two formulas, we see that∆( Y ) = k Y t c (1 − t g Y )1 − t + (1 − t g Y )(1 − t ) P ( t ) . (cid:3) Combining the lemma with the requirement that deg ∆( Y ) < g Y gives Corollary 7.4. If Y is a generalized solid torus, ∆( Y ) ∼ k Y (1 − t g Y ) / (1 − t ) . In contrast, τ ( Y ) is not determined by the fact that Y is a generalized solid torus, as canbe seen by considering the Seifert-fibred spaces M ( ∅ ; a/g, − a/g ). Proposition 7.5. A generalized solid torus is a Floer homology solid torus in the sense ofWatson [18] .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 anappropriate 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 Dehntwist 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 k Y k = g − 2, the criteria of Proposition 3.9 will be satisfied if we take µ = m and λ = l − N m , where N ≫ S µ ⊂ H ( Y ) be the support of \ HF K ( K µ ), normalized so that if x ∈ S µ , then0 ≤ φ ( µ ) ≤ g − S µ is determined by the conditions that for 0 ≤ φ ( µ ) ≤ g − x ∈ S µ ifand only if x ∈ S [ τ ( Y )], and for g − ≤ φ ( µ ) ≤ g − x ∈ S [ µ ] if and only if x − µ S [ τ ( Y )].Similarly, let S λ ⊂ H ( Y ) be the support of \ HF K ( K λ ), normalized so that if x ∈ S λ ,then 0 ≤ φ ( λ ) ≤ ( N + 1) g − S λ is determined by the conditions that for 0 ≤ φ ( x ) ≤ g − x ∈ S λ if and only if x ∈ S [ τ ( Y )], and for g − ≤ φ ( x ) ≤ ( N + 1) g − x ∈ S λ if and onlyif x + λ S [ τ ( Y )]. (Note that φ ( λ ) < 0, so we need x + λ here rather than x − λ ).Now let µ ′ = τ l ( m ) = µ + l and λ ′ = τ l ( λ ) = λ − N l . The supports S µ ′ and S λ ′ can bedescribed similarly.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 withthe fact that φ ( µ ) = g , it is easy to see that f is a bijection. Similarly, if x ∈ S λ , wedefine f to take the unique nonzero element supported at x to the unique nonzero elementof \ 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 ) tothe arrows in the diagram for C ′ = \ CF D ( Y, µ ′ , λ ′ ). Suppose x and y are the initial andterminal ends of an arrow of type D 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 in C ′ .A very similar argument shows that arrows of types D and D are preserved as well. (cid:3) We can prove a partial converse to Proposition 7.1. Recall that Y is said to be semi-primitive 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 > ( Y ) = ∅ , then Y is a generalized solid torus.Proof. Let g = g Y . Since Y is semiprimitive, we have H ( Y ) = Z ⊕ ( Z /g ) and also im ι = g Z ⊕ Z /g ⊂ H ( Y ). Let t, σ be generators of the Z and Z /g summands respectively, so that τ ( Y ) = P ∞ 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 > ( Y ) = 0. We concludethat for a fixed value of k there is at most one value of n for which q k + ng (1) = 0 , g .The Milnor torsion of Y is τ ( Y ) = ∆( Y ) / (1 − t ) = ∞ X i =0 a i t i , where a i = q i (1). Lemma 7.7. There is a constant c so that X 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 welldefined. 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 ) = P ∞ i =0 a i t i . It is easy to see that f ( t ) = 1 + t + . . . + t g − 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 LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 45 implies that ∆( Y ) can be obtained from 1 + t + . . . + t g − by a sequence of operations ofthe first type plus a single operation of the second type, so ∆( Y ) has property (*). (cid:3) 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 as its lowest order term) the subsequence ( a k + ng )has the form . . . , , , , k, g, g, g . . . , where 0 ≤ k ≤ g . In other words, each subsequence isdetermined up to a global shift, and it remains to see how these shifts fit together.We claim that the sequence ( a i ) has the form . . . , , , , , . . . , g − , g, g, g . . . . Equiva-lently, τ ( Y ) ∼ τ = t + 2 t + . . . + ( g − t g − + gt g + gt g +1 + . . . = t (1 − t g )(1 − t ) To see this, let us say that Q ( t ) ∈ Z [ t − , t ]] is obtained from P ( t ) by an elementary shiftif Q ( t ) − P ( t ) = at i + ( g − a ) t i + g for some a, i ∈ Z . We have shown above that τ ( Y ) isobtained from τ by a sequence of elementary shifts.Next, we consider the effect of an elementary shift on the Alexander polynomial. If Q ( t ) ∈ Z [ t − , 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 ) = 1+ . . . + t g − . 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 by a sequenceof elementary shifts and F ( Q ( t )) = F ( τ ), then Q ( t ) is obtained from τ 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 , so Y is a generalized solid torus. (cid:3) As we observed above, if Y is a generalized solid torus, H ( Y, ∂Y ) is generated by asurface of genus 0. It follows that Y ( l ) = Z S × S ), where Z is a rational homologysphere. Conversely, we have Proposition 7.8. Suppose that K ⊂ Z S × S ) has an L-space surgery. Then thecomplement of K is a generalized solid torus.Proof. We use the exact triangle with twisted coefficients, as formulated by Ai and Petersin [1]. We briefly recall their statement. Given a class η ∈ H ( Y ) and µ ∈ Sl ( Y ), we canform ω µ = P D ( j ∗ ( η )) ∈ H ( Y ( µ )), where j : Y → Y ( µ ) is the inclusion. The twisted Floerhomology d HF ( Y ( µ ); Λ ω µ ) is a module over the universal Novikov ringΛ = nX a r t r | r ∈ R , a r ∈ Z , { r < C | a r = 0 } < ∞ for all C ∈ R o . If the image of ω µ in H ( Y ( µ ) , R ) is 0, then d HF ( Y ( µ ); Λ ω µ ) = d HF ( Y ( µ )) ⊗ Λ. Ai andPeters show that if µ · λ = 1, there is a long exact sequence → d HF ( Y ( µ ); Λ ω µ ) → d HF ( Y ( λ ); Λ ω λ ) → d HF ( Y ( µ + λ ); Λ ω µ + λ ) → d HF ( Y ( µ ); Λ ω µ ) → . Let Y be the complement of K , so Y ( l ) = Z S × S ). Choose η ∈ H ( Y ) with φ ( η ) = 1, so that ω l generates H ( Y ( l )) = Z . By [1] Proposition 2.2, d 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 ( Y ( m ); R ) ≃ H ( Y ( m + l ); R ) = 0. The exact triangle shows that d HF ( Y ( m )) ⊗ Λ ≃ d HF ( Y ( m + l )) ⊗ Λ,which implies that d HF ( Y ( m )) ≃ d HF ( Y ( m + l )). Since H ( Y ( m )) ≃ H ( Y ( m + l )), itfollows that m + l ∈ L ( Y ). Repeating, we find that m + nl ∈ L ( Y ) for all n > 0, and thusthat l is a limit point of L ( Y ). It follows that Y is Floer simple and D τ > ( Y ) = ∅ .For the general case, suppose that µ ∈ L ( Y ). Then Y ( l ) is obtained by integer surgeryon K µ K − q/p ⊂ Y ( µ ) L ( q, − p ) for an appropriate choice of p and q . Let Y ′ be thecomplement 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.To conclude the argument, we compute τ ( Y ). Let j : Y → Y ( l ) and j : S × D → Y ( l )be the inclusions. The usual product formula for the torsion says that τ ( Y ( l )) ∼ j ∗ ( τ ( Y )) j ∗ ( τ ( S × D )) . Here τ ( Y ( l )) = τ ( Z S × S )) ∼ | H ( Z ) | (1 − t ) . It is easy to see that the map j ∗ : H ( Y ) /T ors → H ( Y ( l )) /T ors is an isomorphism, whilethe map j ∗ : H ( S × D ) → H ( Y ( l )) /T ors is multiplication by g , so | H ( Z ) | (1 − t ) ∼ τ ( Y )1 − t g . Equivalently τ ( Y ) ∼ | H ( Z ) | − t g (1 − t ) . It follows that Y is a generalized solid torus. (cid:3) Proposition 1.9 from the introduction is an immediate consequence of Propositions 7.6and 7.8, and Proposition 1.11 follows from Proposition 7.5.7.2. NLS Detection. Next, we study the notion of NLS detection introduced by Boyer andClay in [6]. Suppose that Y is a rational homology solid torus and that Y is a semi-primitivegeneralized solid torus. Given a primitive class α ∈ H ( Y ), choose an orientation reversinghomeomorphism ϕ : ∂Y → ∂Y with ϕ ∗ ( α ) = l , where l ∈ H ( ∂Y ) is the homologicallongitude. Since Y is a Floer homology solid torus, d HF ( Y ϕ ) is well defined, in the sensethat any φ satisfying ϕ ∗ ( α ) = l will give the same result. We say that α is NLS detected by Y if Y ϕ is not an L-space.If Y is Floer simple, it follows from Theorem 1.8 that α is NLS detected by Y if andonly if α is not in the interior of L ( Y ). In fact, there is a direct proof of this fact for any Y . Proposition 7.9. The slope α is NLS detected by Y if and only if α is not in the interiorof L ( Y ) .Proof. Suppose that α is not NLS detected by Y . Then Y ϕ i is an L-space for every ϕ i with ϕ i ∗ ( α ) = l . The manifolds Y ϕ i are all obtained by Dehn filling a manifold Y ′ which isconstructed by identifying ν ( α ) ⊂ ∂Y with ν ( l ) ⊂ ∂Y , as in the proof of Lemma 2.7. Itfollows that Y ′ is Floer simple.Let µ ∈ H ( ∂Y ′ ) be the class which represents the common image of α ∈ H ( ∂Y ) and l ∈ H ( ∂Y ). The sutured manifold ( Y ′ , γ µ ) contains an essential annulus A which separates Y from Y . The boundary of A is a pair of curves parallel to µ . We choose the positionof the sutures so that one component of ∂A lies in R + ( γ µ ) and the other component is in R − ( γ µ ). Decomposing ( Y ′ , γ µ ) along A gives a new sutured manifold which is the disjointunion of ( Y , γ α ) and ( Y , γ l ). A is a product annulus, so it follows from Lemma 8.9 of [26]that SF H ( Y ′ , γ µ ) = SF H ( Y , γ α ) ⊗ SF H ( Y , γ l ) . Since ∂Y ′ = T , there is a natural injection c : Spin c ( Y ′ , γ µ ) → H ( Y ′ ) given by theformula j ( s ) = P D ( c ( s )), and similarly for Y and Y . The tensor product respects thedecomposition into Spin c structures in the sense that c ( x ⊗ y ) = j ∗ ( c ( x )) + j ∗ ( c ( y )), where j i : Y i → Y ′ is the inclusion. LOER SIMPLE MANIFOLDS AND L-SPACE INTERVALS 47 In the case at hand, H ( Y ′ ) = H ( Y ) ⊕ H ( Y ) / h α = l i , and H ( Y ) ≃ Z ⊕ Z /g Y , wherethe Z /g Y summand is generated by l . Thus H ( Y ′ ) ≃ Z ⊕ ( H ( Y ) / h g Y α i ). Now α is anontorsion element of H ( Y ) (otherwise Y ϕ is not a rational homology sphere), so the imageof j ∗ is contained in the torsion subgroup of H ( Y ′ ).If Y is a rational homology S × D and β ∈ Sl ( Y ), then SF H ( Y, γ β , s ) = 0 whenever φ ( c ( s )) > k Y k + | φ ( β ) | . The set O ( Y,γ β ) = { s ∈ Spin c ( Y, γ β ) | φ ( c ( s )) = k Y k + | φ ( β ) |} is theset of outer Spin c structures for ( Y, γ β ) [26]. We write SF H ( Y, γ β , O ) = M s ∈ O ( Y,γβ ) SF H ( Y, γ β , s ) . Since the image of j ∗ is contained in the torsion subgroup, we have(154) SF H ( Y ′ , γ µ , O ) ≃ SF H ( Y , γ α ) ⊗ SF H ( Y , γ l , O ) . In particular, k Y ′ k = k Y k = g Y − g Y ′ − 2, so Y ′ is a generalized solid torus.To conclude the proof we use the following two lemmas. The first is probably well-known,but we give a proof just in case. Lemma 7.10. Suppose Y is an incompressible rational homology S × D , that l ∈ H ( ∂Y ) is the homological longitude, and that m · l = 1 . Then SF H ( Y, γ l , O ) ≃ SF H ( Y, γ m , O ) ⊗ H ∗ ( S ) . Proof. Let S ⊂ Y be a properly embedded surface generating H ( Y, ∂Y ). If we decompose( Y, γ m ) along S , we get a sutured manifold ( Z, γ Z ), where ∂Z is a union of two copiesof S glued together their boundaries, and there is one suture for each component of ∂S .Decomposing ( Y, γ l ) along S gives ( Z, γ ′ Z ), where the suture γ ′ Z is the same as γ Z except thatthere are three parallel sutures along one component of ∂S instead of one. By Proposition9.2 of [26], SF H ( Z, γ ′ Z ) ≃ SF H ( Z, γ Z ) ⊗ H ∗ ( S ). (cid:3) Lemma 7.11. If Y is a generalized solid torus and m ∈ H ( ∂Y ) satisfies φ ( m ) = g Y , then SF H ( Y, γ m , O ) ≃ Z k Y .Proof. SF H ( Y, γ m , O ) = \ HF K ( K m , O ), where K m ⊂ Y ( m ) is the dual knot. Since Y isa generalized solid torus, the latter group is Floer simple, hence determined by its Eulercharacteristic. By Lemma 7.3, φ ( χ ( \ HF K ( K m ))) = k Y (1 − t g Y ) (1 − t ) . It follows that \ HF K ( K m , O ) ≃ Z k Y . (cid:3) Applying the lemmas to Y , which has k Y = 1, we see that SF H ( Y , γ l , O ) ≃ H ∗ ( S ).For Y ′ , suppose that H ( Y ( α )) = H ( Y ) / h α i has order d . The torsion subgroup of H ( Y ′ )is H ( Y ) / h g Y α i , so it has order g Y d , Since g Y ′ = g Y , we see that k Y ′ = d . Since µ is the homological longitude of Y ′ , SF H ( Y ′ , γ µ , O ) ≃ H ∗ ( S ) ⊗ Z d . Comparing withequation (154), we see that SF H ( Y , γ α ) ≃ Z d . Now if K α ⊂ Y ( α ) is the dual knot, then \ HF K ( K α ) = SF H ( Y , γ α ) ≃ Z d , where d = | H ( Y ( α )) | . So K α is Floer simple, whichimplies that Y is Floer simple and that α is in the interior of L ( Y ).Conversely, if α is in the interior of L ( Y ), Theorem 1.8 implies that Y ϕ is an L-space, so α is not NLS detected by Y . (cid:3) Boyer and Clay define α to be NLS detected if it is NLS detected by some N g , where N g = M (1 /g, − /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 detectedby 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. Someof these were previously known to Hanselman and Watson [18] and Vafaee [46]. We startwith the following observation. Corollary 7.12. If Y is an irreducible, semi-primitive generalized solid torus, then Y isthe complement of a closed g Y -strand braid in S × S .Proof. The hypotheses imply that ∆( Y ) ∼ (1 − t g ) / (1 − t ) and that H ( Y, ∂Y ) is generatedby a g Y -times punctured sphere. By Corollary 2.3, it follows that Y fibres over S withfibre of genus 0. (cid:3) By Proposition 7.8, any knot in S × S with a lens space surgery is a generalized solidtorus. Cebanu [10] showed that a knot of this form is a closed braid in S . Examples ofsuch knots were studied by Buck, Baker, and Leucona in [2]. Many (but not all) of themare derived from knots in the solid torus which have solid torus surgeries. These knots werecompletely classified by Gabai [15] and Berge [3].To find other examples, we look for braids in S × S which have L-space surgeries. Onecriterion for finding such examples is given here. Suppose σ is an ordinary g strand braid in D × I . We can close σ to get a closed braid in S × D . Dehn filling S × D along S × p gives the ordinary braid closure σ ⊂ S . We can also fill S × D along ∂D to get a closedbraid in S × S , which we denote by e 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 L-space knot in S for all n ≥ . Then the complement of e σ is a semi-primitive generalizedsolid torus.Proof. Let L ⊂ S 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 × S 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, − /n ) is the result of a + ng surgery on K n . UsingSeifert’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 − / 2. Thus if a > C ( σ ), then a + ng ≥ g ( K n ) − n ≥ K n is a positive L-space knot, so L ( a, − /n ) is an L-space for all n > Y be the manifold obtained by doing a surgery on K , and let Y = Y − ν ( B ).There is a slope α ∈ Sl ( Y ) so that Y ( α ) = L ( a, α − /n ∈ Sl ( Y )which converge to α such that Y ( α − /n ) = L ( a, − /n ). It follows that Y is Floer simpleand that α is in the closure of L ( Y ). Since α is not the homological longitude of Y , α ∈ L ( Y ), so L ( a, 0) is an L-space. By Proposition 7.8, Y is a generalized solid torus. 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