aa r X i v : . [ m a t h . G T ] O c t INSTANTONS AND L-SPACE SURGERIES
JOHN A. BALDWIN AND STEVEN SIVEK
Abstract.
We prove that instanton L-space knots are fibered and strongly quasipositive.Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homol-ogy, and includes a new decomposition theorem for cobordism maps in framed instantonFloer homology akin to the Spin c decompositions of cobordism maps in other Floer homol-ogy theories. As our main application, we prove (modulo a mild nondegeneracy condition)that for r a positive rational number and K a nontrivial knot in the 3-sphere, there existsan irreducible homomorphism π ( S r ( K )) → SU (2)unless r ≥ g ( K ) − K is both fibered and strongly quasipositive, broadly generalizingresults of Kronheimer and Mrowka. We also answer a question of theirs from 2004, provingthat there is always an irreducible homomorphism from the fundamental group of 4-surgeryon a nontrivial knot to SU (2). In another application, we show that a slight enhancementof the A-polynomial detects infinitely many torus knots, including the trefoil. Introduction
The most important invariant of a 3-manifold is its fundamental group. One of the mostfruitful approaches to understanding the fundamental group is to study its homomorphismsinto simpler groups. SU (2) is an especially convenient choice because it is one of the simplestnonabelian Lie groups, and because gauge theory provides powerful tools for studying SU (2)representations of 3-manifold groups. Given a 3-manifold Y , we will therefore be interestedin the representation variety(1.1) R ( Y ) = Hom( π ( Y ) , SU (2)) . If this variety is to tell us anything about the 3-manifold not captured by its first homology,it must contain elements with nonabelian image. Let us introduce the following terminologyfor when this is not the case.
Definition 1.1.
A 3-manifold Y is SU (2) -abelian if every ρ ∈ R ( Y ) has abelian image. SU (2)-abelian manifolds can be thought of as the simplest manifolds from the perspectiveof the variety (1.1). A basic question when studying representation varieties is whether theycontain irreducible homomorphisms; since a homomorphism into SU (2) is reducible iff it isabelian, this question admits the following satisfying answer for the varieties studied here. Remark 1.2. R ( Y ) contains an irreducible iff Y is not SU (2)-abelian.Little is known as to which 3-manifolds are SU (2)-abelian, even among Dehn surgeries onknots in S . One of the first major advances in this direction was Kronheimer and Mrowka’slandmark proof of the Property P conjecture [KM04b], in which they showed that S ( K ) is If b ( Y ) = 0 then ρ ∈ R ( Y ) has abelian image iff it has cyclic image. not SU (2)-abelian (and hence not a homotopy 3-sphere) for any nontrivial knot K . Theythen substantially strengthened this result as follows. Theorem 1.3 ([KM04a]) . Suppose K ⊂ S is a nontrivial knot. Then S r ( K ) is not SU (2) -abelian for any rational number r with | r | ≤ . It is natural to ask whether Theorem 1.3 holds for other values of r . Bearing in mind that5-surgery on the right-handed trefoil is a lens space, and hence SU (2)-abelian, Kronheimerand Mrowka posed this question for the next two integer values of r . Question 1.4 ([KM04a]) . Suppose K ⊂ S is a nontrivial knot. Can S ( K ) or S ( K ) be SU (2) -abelian? In this paper, we use instanton Floer homology to prove the following broad generalization(modulo a mild nondegeneracy condition) of Theorem 1.3. In particular, this generalizationestablishes a link between the genus, the smooth slice genus, and the SU (2) representationvarieties of Dehn surgeries on a knot. Theorem 1.5.
Suppose K ⊂ S is a nontrivial knot, and r = mn > is a rational numbersuch that ∆ K ( ζ ) = 0 for any m th root of unity ζ . Then S r ( K ) is not SU (2) -abelian unless r ≥ g ( K ) − and K is both fibered and strongly quasipositive. Remark 1.6.
As alluded to above, the assumption on the Alexander polynomial in The-orem 1.5 is equivalent to a certain nondegeneracy condition (see § m is a prime power. Note that rationals with prime power numeratorsare dense in the reals (see Remark 9.3). Remark 1.7.
The right-handed trefoil is the only fibered, strongly quasipositive knot ofgenus 1. Since r -surgery on this trefoil is not SU (2)-abelian for any r ∈ [0 , r with prime power numerators; e.g., for r = 1 , Theorem 1.8.
Suppose K ⊂ S is a nontrivial knot of genus g . Then • S ( K ) is not SU (2) -abelian, • S r ( K ) is not SU (2) -abelian for any r = mn ∈ (2 , with m a prime power, and • S ( K ) is not SU (2) -abelian unless K is fibered and strongly quasipositive and g = 2 . Remark 1.9.
We expect that S ( K ) is not SU (2)-abelian for any nontrivial knot but donot know how to prove this at present.We highlight two additional applications of our work below before describing the gauge-theoretic results underpinning these theorems.First, in [SZ17], Sivek and Zentner explored the question of which knots in S are SU (2) -averse , meaning that they are nontrivial and have infinitely many SU (2)-abelian surgeries. NSTANTONS AND L-SPACE SURGERIES 3
These include torus knots, which have infinitely many lens space surgeries [Mos71], andconjecturally nothing else. Below, we provide strong new restrictions on SU (2)-averse knotsand their limit slopes , defined for an SU (2)-averse knot as the unique accumulation pointof its SU (2)-abelian surgery slopes. Theorem 1.10.
Suppose K ⊂ S is an SU (2) -averse knot. Then K is fibered, and either K or its mirror is strongly quasipositive with limit slope strictly greater than g ( K ) − . Finally, one of the first applications of Theorem 1.3 was a proof by Dunfield–Garoufalidis[DG04] and Boyer–Zhang [BZ05] that the A-polynomial [CCG +
94] detects the unknot. TheA-polynomial A K ( M, L ) ∈ Z [ M ± , L ± ]is defined in terms of the SL (2 , C ) character variety of π ( S \ K ), and the key idea behindthese proofs is that Theorem 1.3 implicitly provides, for a nontrivial knot K , infinitely manyrepresentations π ( S \ K ) → SU (2) ֒ → SL (2 , C )whose characters are distinct, even after restricting to the peripheral subgroup.More recently, Ni and Zhang proved [NZ17] that ˜ A K ( M, L ) together with knot Floer ho-mology detects all torus knots, where the former is a slight enhancement of the A-polynomialwhose definition omits the curve of reducible characters in the SL (2 , C ) character variety(see § A -polynomial nor its enhancement (nor knot Floer homology) alonecan detect all torus knots. Nevertheless, we use Theorem 1.10, together with an argumentinspired by the proofs of the unknot detection result above, to prove that this enhanced A -polynomial suffices in infinitely many cases, as follows. Theorem 1.11.
Suppose K is either a trefoil or a torus knot T p,q where p and q are distinctodd primes. Then ˜ A J ( M, L ) = ˜ A K ( M, L ) iff J is isotopic to K . Remark 1.12.
We prove that ˜ A K ( M, L ) detects many other torus knots as well; see Corol-lary 10.16 for a more complete list.1.1.
Instanton L-spaces.
The theorems above rely on new results about framed instantonhomology , defined by Kronheimer and Mrowka in [KM11b] and developed further by Scadutoin [Sca15]. This theory associates to a closed, oriented 3-manifold Y and a closed, embeddedmulticurve λ ⊂ Y a Z / Z -graded C -module I ( Y, λ ) = I ( Y, λ ) ⊕ I ( Y, λ ) . This module depends, up to isomorphism, only on Y and the homology class[ λ ] ∈ H ( Y ; Z / Z ) . Let us write I ( Y ) to denote I ( Y, λ ) with [ λ ] = 0. Scaduto proved [Sca15] that if b ( Y ) = 0then I ( Y ) has Euler characteristic | H ( Y ; Z ) | , which implies thatdim I ( Y ) ≥ | H ( Y ; Z ) | . This inspires the following terminology, by analogy with Heegaard Floer homology.
JOHN A. BALDWIN AND STEVEN SIVEK
Definition 1.13.
A rational homology 3-sphere Y is an instanton L-space ifdim I ( Y ) = | H ( Y ; Z ) | . A knot K ⊂ S is an instanton L-space knot if S r ( K ) is an instanton L-space for somerational number r > Remark 1.14.
It follows easily from the discussion above that a rational homology 3-sphere Y is an instanton L-space iff I ( Y ) = 0.Our main Floer-theoretic result is the following. Theorem 1.15.
Suppose K ⊂ S is a nontrivial instanton L-space knot. Then • K is fibered and strongly quasipositive, and • S r ( K ) is an instanton L-space for a rational number r iff r ≥ g ( K ) − . Let us explain how Theorem 1.15 bears on the results claimed in the previous section.The connection with SU (2) representations comes from the principle that I ( Y ) shouldbe the Morse–Bott homology of a Chern–Simons functional with critical setCrit( CS ) ∼ = R ( Y ) . This heuristic holds true so long as the elements in R ( Y ) are nondegenerate in the Morse–Bott sense. In [BS18], we observed that if Y is SU (2)-abelian and b ( Y ) = 0, then H ∗ ( R ( Y ); C ) ∼ = C | H ( Y ; Z ) | , which then implies that Y is an instanton L -space if all elements of R ( Y ) are nondegenerate.For SU (2)-abelian Dehn surgeries, this nondegeneracy is equivalent to the condition on theAlexander polynomial in Theorem 1.5, by work of Boyer and Nicas [BN90]; see [BS18, § Proof of Theorem 1.5.
Suppose K ⊂ S is a nontrivial knot, and r = mn > K ( ζ ) = 0 for any m th root of unity ζ . Suppose S r ( K ) is SU (2)-abelian.Then S r ( K ) is an instanton L-space, by [BS18, Corollary 4.8]. Theorem 1.15 then impliesthat K is fibered and strongly quasipositive, and r ≥ g ( K ) − (cid:3) Proof of Theorem 1.10.
Suppose K ⊂ S is SU (2)-averse with limit slope r >
0. Then K has an instanton L-space surgery of slope ⌈ r ⌉ − >
0, by [SZ17, Theorem 1.1]. As K is nontrivial by definition, Theorem 1.15 then implies that K is fibered and stronglyquasipositive, and r > ⌈ r ⌉ − ≥ g ( K ) − . If r <
0, then we apply the same argument to the mirror K , whose limit slope is − r > (cid:3) Theorem 1.15 will be unsurprising to those familiar with Heegaard Floer homology, giventhe conjectural isomorphism [KM10, Conjecture 7.24] I ( Y ) ∼ = conj . d HF ( Y ) ⊗ C , and the fact that nontrivial knots with positive Heegaard Floer L-space surgeries are alreadyknown to be fibered [OS05b, Ni07] and strongly quasipositive [Hed10], with L-space surgeryslopes comprising [OS11] [2 g ( K ) − , ∞ ) ∩ Q . NSTANTONS AND L-SPACE SURGERIES 5
On the other hand, all proofs in the literature of the fiberedness result alone use some subsetof: the Z ⊕ Z -filtered Heegaard Floer complex associated to a knot, the large surgery formula,the ( ∞ , , n )-surgery exact triangle for n >
1, and, in all cases, the Spin c decomposition ofthe Heegaard Floer groups of rational homology spheres. None of this structure is availablein framed instanton homology.Our proof of Theorem 1.15 is thus, by necessity, largely novel (and can even be translatedto Heegaard Floer and monopole Floer homology to give new, conceptually simpler proofsof the analogous theorems in those settings). One of the ingredients is a new decompositiontheorem for cobordism maps in framed instanton homology, analogous to the Spin c decom-positions of cobordism maps in Heegaard and monopole Floer homology, described in thenext section. We expect this decomposition result to have other applications as well.1.2. A decomposition of framed instanton homology.
The framed instanton homol-ogy of a 3-manifold comes equipped with a collection of commuting operators µ : H ( Y ; Z ) → End( I ( Y, λ ))such that the eigenvalues of µ ([Σ]) are even integers between 2 − g (Σ) and 2 g (Σ) − ⊂ Y . Following Kronheimer and Mrowka [KM10, Corollary 7.6], this gives riseto an eigenspace decomposition I ( Y, λ ) = M s : H ( Y ; Z ) → Z I ( Y, λ ; s ) , where each summand I ( Y, λ ; s ) is the simultaneous generalized s ( h )-eigenspace of µ ( h ) forall h ∈ H ( Y ; Z ). Only finitely many of these summands are nonzero.A smooth cobordism ( X, ν ) : ( Y , λ ) → ( Y , λ )induces a homomorphism(1.2) I ( X, ν ) : I ( Y , λ ) → I ( Y , λ ) . We extend the eigenspace decomposition discussed above to cobordism maps, in a way whichmirrors the Spin c decomposition of cobordism maps in, say, the hat flavor of Heegaard Floerhomology. Our main theorem in this vein is Theorem 1.16 below. In stating it, we will usethe following notation: given an inclusion i : M ֒ → N of two manifolds and a homomorphism s : H ( N ; Z ) → Z , we write s | M = s ◦ i ∗ : H ( M ; Z ) → H ( N ; Z ) → Z for the restriction of s to M . Theorem 1.16.
Let ( X, ν ) : ( Y , λ ) → ( Y , λ ) be a cobordism with b ( X ) = 0 . Then thereis a natural decomposition of the cobordism map (1.2) into a sum I ( X, ν ) = X s : H ( X ; Z ) → Z I ( X, ν ; s ) of maps of the form I ( X, ν ; s ) : I ( Y , λ ; s | Y ) → I ( Y , λ ; s | Y ) The analogue of Theorem 1.15 is known to hold in monopole Floer homology, but only by appeal to theisomorphism between monopole Floer homology and Heegaard Floer homology.
JOHN A. BALDWIN AND STEVEN SIVEK with the following properties.(1) I ( X, ν ; s ) = 0 for all but finitely many s .(2) If I ( X, ν ; s ) is nonzero, then s ( h ) + h · h ≡ for all h ∈ H ( X ; Z ) , and s satisfies an adjunction inequality | s ([Σ]) | + [Σ] · [Σ] ≤ g (Σ) − for every smoothly embedded, connected surface Σ ⊂ X of genus at least andpositive self-intersection.(3) If ( X, ν ) is a composition of two cobordisms ( Y , λ ) ( X ,ν ) −−−−−−→ ( Y , λ ) ( X ,ν ) −−−−−−→ ( Y , λ ) where b ( X ) = b ( X ) = 0 , then we have a composition law I ( X , ν ; s ) ◦ I ( X , ν ; s ) = X s : H ( X ; Z ) → Z s | X = s s | X = s I ( X, ν ; s ) for all s : H ( X ; Z ) → Z and s : H ( X ; Z ) → Z .(4) If ˜ X = X CP denotes the blow-up of X , with exceptional class E , then I ( ˜ X, ν ; s + kE ) = ( I ( X, ν ; s ) k = ± k = ± for all s : H ( X ; Z ) → Z and all k ∈ Z .(5) I ( X, ν + α ; s ) = ( − ( s ( α )+ α · α )+ ν · α I ( X, ν ; s ) for all α ∈ H ( X ; Z ) . The key to our proof of this theorem is a cobordism analogue of Kronheimer and Mrowka’sstructure theorem [KM95a] for the Donaldson invariants of closed 4-manifolds. Briefly, onecan extend the cobordism map (1.2) to maps D X,ν : I ( Y , λ ) ⊗ A ( X ) → I ( Y , λ ) , where A ( X ) is the graded algebra A ( X ) = Sym( H ( X ; R ) ⊕ H ( X ; R )) , such that I ( X, ν ) = D X,ν ( − ⊗ . Letting x = [pt] in H ( X ; Z ), one defines a formal power series D νX ( h ) = D X,ν (cid:16) − ⊗ (cid:16) e h + xe h (cid:17)(cid:17) for each h ∈ H ( X ; R ), following [KM95a], and we prove that this series can be expressedas a finite sum e Q X ( h ) / r X j =1 a j e K j ( h ) , where the K j are basic classes , which we think of as homomorphisms H ( X ; Z ) → Z . Thishas exactly the same form as Kronheimer and Mrowka’s original structure theorem, exceptthat the coefficients are now homomorphisms a j : I ( Y , λ ) → I ( Y , λ ) NSTANTONS AND L-SPACE SURGERIES 7 instead of rational numbers. Up to scaling, we define I ( X, ν ; K j ) to be the map a j .To prove this structure theorem for cobordisms, we adapt Mu˜noz’s [Mu˜n00] alternativeproof of Kronheimer and Mrowka’s structure theorem for closed 4-manifolds. The proofs ofthat theorem in both [KM95a] and [Mu˜n00] require that the 4-manifold contains a surface ofpositive self-intersection. The cobordism analogue in the case b +2 ( X ) = 0 therefore requiresa new argument (see § On the proof of Theorem 1.15.
We provide below a detailed sketch of the proof ofour main Floer-theoretic result, Theorem 1.15.Our proof that instanton L-space knots are fibered has two main components. The firstis the result that the framed instanton homology of 0-surgery on a knot detects fiberedness.For this, suppose Σ is a genus-minimizing Seifert surface for a nontrivial knot K ⊂ S withmeridian µ . Let ˆΣ denote the capped-off surface in S ( K ), and let s i : H ( S ( K )) → Z be the homomorphism defined by s i ([ ˆΣ]) = 2 i , for each i ∈ Z . We prove the following,largely using a combination of arguments due to Kronheimer and Mrowka [KM10] and Ni[Ni08]; this result may be of independent interest. Theorem 1.17. If K ⊂ S is a nontrivial knot as above, then dim I ( S ( K ) , µ ; s g ( K ) − ) ≥ , with equality iff K is fibered. The second component, which is conceptually novel, uses only the surgery exact triangleand formal properties of the decomposition of cobordism maps given by Theorem 1.16, asfollows. For each k ≥
0, there is a surgery exact triangle · · · → I ( S ) I ( X k ,ν k ) −−−−−−→ I ( S k ( K )) I ( W k +1 ,ω k +1 ) −−−−−−−−−−→ I ( S k +1 ( K )) → . . . , where we use I ( S ( K ) , µ ) instead of I ( S ( K )) when k = 0. Here, X k is the trace of k -surgery on K , and W k +1 is the trace of − K in S k ( K ), so that(1.3) X k ∪ S k ( K ) W k +1 ∼ = X k +1 CP . We apply (1.3) inductively, using the blow-up formula and adjunction inequality in Theorem1.16, to eventually prove (Lemma 7.5) that, for any n >
0, the kernel of the composition(1.4) I ( S ( K ) , µ ) I ( W ,ω ) −−−−−−−→ I ( S ( K )) I ( W ,ω ) −−−−−−−→ . . . I ( W n ,ω n ) −−−−−−−→ I ( S n ( K ))lies in the subspace of I ( S ( K ) , µ ) spanned by the elements y i = I ( X , ν ; t ,i )( ) ∈ I ( S ( K ) , µ ; s i ) , for i = 1 − g ( K ) , − g ( K ) , . . . , g ( K ) − , where each homomorphism t ,i : H ( X ) → Z is defined by t ,i ([ ˆΣ]) = 2 i , and generates I ( S ) ∼ = C . Noting that the composition (1.4)preserves the Z / Z -grading, it follows that if S n ( K ) is an instanton L-space then(1.5) dim I ( S ( K ) , µ ; s g ( K ) − ) = 1 , JOHN A. BALDWIN AND STEVEN SIVEK which then implies that K is fibered by Theorem 1.17. Indeed, if (1.5) does not hold, thenthe group I ( S ( K ) , µ ; s g ( K ) − )contains an element which is not a multiple of y g ( K ) − , which is thus sent by the composition(1.4) to a nonzero element of I ( S n ( K )) = 0(see Remark 1.14), a contradiction. The same reasoning shows that y g ( K ) − = 0 when K isan instanton L-space knot.For strong quasipositivity, suppose K is an instanton L-space knot. Since K is fibered, itsupports some contact structure ξ K on S . Then K is strongly quasipositive iff ξ K is tight[Hed10, BI09]. To prove that this contact structure is tight, we first recall from above that y g ( K ) − = 0. By duality, this implies that the map(1.6) I ( − S ( K ) , µ ; s − g ( K ) ) ∼ = C → I ( − S )induced by turning X upside down has nonzero image. The image of the analogous map inHeegaard Floer homology defines the Heegaard Floer contact invariant c ( ξ K ) [OS05a], and c ( ξ K ) = 0 implies that ξ K is tight. Unfortunately, we cannot prove that the nontrivialityof (1.6) implies the tightness of ξ K in general (the Heegaard Floer proof ultimately usesthe description of c ( ξ K ) in terms of the knot Floer filtration for K , for which there is noanalogue in the instanton setting).Our solution involves cabling. Namely, for coprime integers p and q with q ≥ pq > g ( K ) − , a simple argument shows that the ( p, q )-cable K p,q is also an instanton L-space knot, whichimplies as above that the map I ( − S ( K p,q ) , µ ; s − g ( K p,q ) ) → I ( − S )is nonzero. Using the fact that such cables can be deplumbed, we show that the nontrivialityof this map implies that a slight variant of the contact class we defined in [BS16] is nonzerofor the contact structure ξ K ,q corresponding to the cable K ,q . We ultimately deduce fromthis that ξ K ,q is tight (we only work this out concretely in the case q = 2, which suffices forour application). Noting that ξ K ∼ = ξ K ,q for any positive q , we conclude that ξ K is tight aswell.Finally, our characterization of the L-space surgery slopes has two parts. First, we provedin [BS18, Theorem 4.20] that if the set of positive instanton L-space slopes for a nontrivialknot K is nonempty, then it has the form[ N, ∞ ) ∩ Q for some positive integer N , and we show in Proposition 7.11 that(1.7) N ≤ g ( K ) − . To prove that this inequality is actually an equality , we then employ an observation dueto Lidman, Pinz´on-Caicedo, and Scaduto in [LPCS19]. There, they use the Z / Z -gradingof our contact invariant [BS16] together with our main technical result in [BS18] to prove, NSTANTONS AND L-SPACE SURGERIES 9 under the assumptions that K is a nontrivial instanton L-space knot with maximal self-linking number(1.8) sl ( K ) = 2 g ( K ) − , that(1.9) N ≥ g ( K ) − § N = 2 g ( K ) − all nontrivial instanton L-space knots, as desired. Remark 1.18.
Some of the arguments in our proof of Theorem 1.15 invoke the Girouxcorrespondence [Gir02] between contact structures and open books. This correspondence isgenerally accepted to be true, but since a complete proof has yet to appear in the literature,we have chosen to indicate below which of our results depend on it.We use the Giroux correspondence to prove the strong quasipositivity claim in Theo-rem 1.15 (and therefore in Theorems 1.5 and 1.10 as well), which is then used to prove theclaim in Theorem 1.15 that positive instanton L-space slopes r satisfy the bound r ≥ g ( K ) − . This bound is required for the 3- and 4-surgery cases of Theorem 1.8, and in Theorem 1.11 toprove that ˜ A K ( M, L ) detects any of the claimed torus knots other than the trefoils, which arehandled separately by Theorem 10.14. We remark that Theorem 7.8 and Proposition 7.12do not rely on this correspondence. These assert that an instanton L-space knot is fiberedwith genus equal to its smooth slice genus; and that if S ( K ) or S ( K ) is an instantonL-space, then K is either the unknot or the right-handed trefoil.1.4. Organization.
We provide some background on instanton Floer homology in §
2. Thematerial in § § §
5, we prove the structure theorem for cobordism maps on framed instanton homology,which we then use in § §
7, we prove thatinstanton L-space knots are fibered with Seifert genus equal to smooth slice genus (Theo-rem 7.8). We prove in § §
9, we inves-tigate SU (2)-abelian surgeries with small surgery slope, and prove Theorem 1.8. We studythe A-polynomials of torus knots in §
10, and prove Theorem 1.11.1.5.
Acknowledgments.
We thank Ken Baker, John Etnyre, Matt Hedden, Jen Hom, andTom Mrowka for helpful conversations. We thank Tye Lidman, Juanita Pinz´on-Caicedo,and Chris Scaduto for the same, and also for discussing their work [LPCS19] with us whileit was still in progress. JAB was supported by NSF CAREER Grant DMS-1454865. Background
In this section, we provide background on instanton Floer homology and establish somenotational conventions. Much of our discussion below is adapted from [KM10] and [Sca15].2.1.
Conventions.
All manifolds in this paper are smooth, oriented, and compact, and allsubmanifolds are smoothly and properly embedded.2.2.
Instanton Floer homology.
Let (
Y, λ ) be an admissible pair , meaning that Y is aclosed, connected 3-manifold, and λ ⊂ Y is a multicurve which intersects some surface inan odd number of points. We associate to this pair: • a Hermitian line bundle w → Y with c ( w ) Poincar´e dual to λ , and • a U (2) bundle E → Y equipped with an isomorphism θ : ∧ E → w .The instanton Floer homology I ∗ ( Y ) λ is roughly the Morse homology, with C -coefficients,of the Chern–Simons functional on the space B = C / G of SO (3)-connections on ad( E ) modulo determinant-1 gauge transformations (the automor-phisms of E which respect θ ), as in [Don02, § Z / Z -grading,which reduces to a canonical Z / Z -grading [Frø02, Don02]. Remark 2.1.
Up to isomorphism, the group I ∗ ( Y ) λ depends only on Y and the homologyclass [ λ ] ∈ H ( Y ; Z / Z )of the multicurve λ .Each homology class h ∈ H k ( Y ; R ) gives rise to a class µ ( h ) ∈ H − k ( B ), and thereforeto an endomorphism µ ( h ) : I ∗ ( Y ) λ → I ∗ + k − ( Y ) λ of degree k −
4, as in [DK90]. These endomorphisms are additive in the sense that µ ( h + h ) = µ ( h ) + µ ( h )for h , h ∈ H k ( Y ; R ), and endomorphisms associated to even-dimensional classes commute.This implies that for any collection of even-dimensional classes, I ∗ ( Y ) λ is the direct sum ofthe simultaneous generalized eigenspaces of the associated operators. Remark 2.2.
Going forward, we will often blur the distinction between closed submanifoldsof Y and the homology classes they represent, and we will use eigenspace to mean generalizedeigenspace unless stated otherwise.After choosing an absolute lift of the relative Z / Z -grading on I ∗ ( Y ) λ , we can write anelement of this group as v = ( v , v , v , v , v , v , v , v ) , where v i is in grading i ∈ Z / Z . Let φ : I ∗ ( Y ) λ → I ∗ ( Y ) λ be the map defined for v as above by(2.1) φ ( v ) = ( v , v , iv , iv , − v , − v , − iv , − iv ) . This map gives rise to isomorphisms between eigenspaces, as below.
NSTANTONS AND L-SPACE SURGERIES 11
Lemma 2.3.
For any h ∈ H ( Y ; R ) and m, n ∈ C , the map φ defines an isomorphism fromthe ( m, n ) -eigenspace of the operators µ ( h ) , µ (pt) on I ∗ ( Y ) λ to the ( im, − n ) -eigenspace.Proof. This follows easily from the fact that µ ( α ) has degree − µ (pt) degree − (cid:3) Using work of Mu˜noz [Mu˜n99b], Kronheimer and Mrowka prove the following in [KM10,Corollary 7.2 & Proposition 7.5].
Theorem 2.4.
Let ( Y, λ ) be an admissible pair, and suppose R ⊂ Y is a connected surfaceof genus g > . Then the simultaneous eigenvalues of the operators µ ( R ) , µ (pt) on I ∗ ( Y ) λ are contained in the set { ( i r · k, ( − r · } , for ≤ r ≤ and ≤ k ≤ g − . This theorem implies that µ (pt) − I ∗ ( Y ) λ , but in some cases we knowthat it is in fact the zero operator, per the following result of Frøyshov [Frø02, Theorem 9](the result stated below uses the observation “ N = N = 1” preceding the cited theorem,which follows from [Mu˜n99b]). Theorem 2.5. If ( Y, λ ) is an admissible pair such that λ intersects a surface of genus atmost in an odd number of points, then µ (pt) − ≡ on I ∗ ( Y ) λ . Theorem 2.4 motivates the definition below.
Definition 2.6.
Let (
Y, λ ) be an admissible pair, and R ⊂ Y a disjoint union R = R ∪ · · · ∪ R n , where each R i is a connected surface in Y of genus g i >
0. We define I ∗ ( Y | R ) λ ⊂ I ∗ ( Y ) λ to be the (2 g − , . . . , g n − , µ ( R ) , . . . , µ ( R n ) , µ (pt). Remark 2.7.
Note that the Z / Z -grading on I ∗ ( Y ) λ descends to a canonical Z / Z -gradingon I ∗ ( Y | R ) λ since the operators µ ( R i ) and µ (pt) have even degree.We extend this definition to disconnected manifolds via tensor product (we will implicitlyuse this more general definition in the statement of Theorem 2.12). Remark 2.8.
For any connected surface Σ ⊂ Y of genus g >
0, the eigenvalues of µ (Σ)acting on I ∗ ( Y | R ) λ are contained in the set of even integers { − g, − g, . . . , g − } , by Theorem 2.4 and the fact that I ∗ ( Y | R ) λ is contained in the 2-eigenspace of µ (pt). Wewill therefore refer to the (2 g − µ (Σ) as the top eigenspace of this operator. Example 2.9.
It follows from [Mu˜n99b] that I ∗ ( S × R | R ) γ ∼ = C for R a connected surface of positive genus, and γ = S × { pt } .We will make extensive use of the following nontriviality result due to Kronheimer andMrowka [KM10, Theorem 7.21]. Theorem 2.10.
Let ( Y, λ ) be an admissible pair with Y irreducible, and suppose R ⊂ Y is a connected surface which minimizes genus in its homology class, such that λ · R is odd.Then I ∗ ( Y | R ) λ is nonzero. Cobordism maps.
Suppose ( Y , λ ) and ( Y , λ ) are admissible pairs, and let ( X, ν )be a cobordism from the first pair to the second. We associate to this cobordism: • a Hermitian line bundle w → X with c ( w ) Poincar´e dual to ν , and • a U (2) bundle E → X equipped with an isomorphism θ : ∧ E → w which restrict to the bundle data associated to the admissible pairs at either end. One thendefines a map I ∗ ( X ) ν : I ∗ ( Y ) λ → I ∗ ( Y ) λ in the standard way: given generators a i ∈ I ∗ ( Y i ) λ i , one considers an associated configura-tion space B ( X, E, a , a )of SU (2)-connections on E modulo determinant-1 gauge transformations, and defines thecoefficient h I ∗ ( X ) ν ( a ) , a i to be a count of projectively anti-self-dual instantons in a 0-dimensional moduli space M ( X, E, a , a ) ⊂ B ( X, E, a , a ) . The map I ∗ ( X ) ν is homogeneous with respect to the Z / Z grading, and shifts this gradingby the amount below, according to [Sca15, § § I ∗ ( X ) ν ) = −
32 ( χ ( X ) + σ ( X )) + 12 ( b ( Y ) − b ( Y )) (mod 2) . More generally, one can define a mapΨ
X,ν : I ∗ ( Y ) λ ⊗ A ( X ) → I ∗ ( Y ) λ , where A ( X ) is the symmetric graded algebra on H ( X ; R ) ⊕ H ( X ; R )in which H k ( X ; R ) has grading 4 − k . Namely, a monomial z = c c . . . c k ∈ A ( X ) of degree d gives rise to a class µ ( c ) ∪ µ ( c ) ∪ · · · ∪ µ ( c k ) ∈ H d ( B ( X, E, α , α )) , following [DK90], and the coefficient h Ψ X,ν ( a ⊗ z ) , a i is the sum of the evaluations of this class on the d -dimensional components of M d ( X, E, a , a ).In particular, I ∗ ( X ) ν = Ψ X,ν ( − ⊗ . Remark 2.11.
The maps Ψ
X,ν are well-defined up to sign, and depend only on X and thehomology class [ ν ] ∈ H ( X, ∂X ; Z / Z ) . We will ignore this sign ambiguity, which can be removed by choosing a homology orientationon X , as in [KM11b, § NSTANTONS AND L-SPACE SURGERIES 13
The map Ψ
X,ν interacts nicely with the actions of H ∗ ( Y i ; R ) on I ∗ ( Y i ) λ i described in § h i ∈ H ∗ ( Y i ; R ) for i = 0 ,
1, we have the relationsΨ
X,ν ( µ ( h ) a ⊗ z ) = Ψ X,ν ( a ⊗ h z ) , (2.3) µ ( h )Ψ X,ν ( a ⊗ z ) = Ψ X,ν ( a ⊗ h z ) . (2.4)Therefore, if h and h are homologous in X , we haveΨ X,ν ( µ ( h ) − ⊗ − ) = µ ( h )Ψ X,ν ( − ⊗ − ) . In particular, the cobordism map I ∗ ( X ) ν intertwines the actions of µ ( h ) and µ ( h ) in thiscase, and therefore respects the corresponding eigenspace decompositions. It follows that if R ⊂ Y and R ⊂ Y are surfaces of the same positive genus which are homologous in X then I ∗ ( X ) ν restricts to a map I ∗ ( X ) ν : I ∗ ( Y | R ) λ → I ∗ ( Y | R ) λ . The excision theorem.
Let Y be a closed 3-manifold, Σ ∪ Σ ⊂ Y a disjoint unionof connected surfaces of the same positive genus, and λ a multicurve in Y which intersectseach of Σ and Σ in the same odd number of points with the same sign, λ ∩ Σ ) = | λ · Σ = λ · Σ | = λ ∩ Σ ) , according to the following two cases: • if Y is connected, we require that Σ is not homologous to Σ ; • if Y has two components, we require that one Σ i is contained in each component.Let ϕ : Σ → Σ be an orientation-reversing diffeomorphism such that ϕ ( λ ∩ Σ ) = ϕ ( λ ∩ Σ ) . Let Y ′ be the manifold with four boundary components, ∂Y ′ = Σ ∪ − Σ ∪ Σ ∪ − Σ , obtained from Y by cutting along Σ ∪ Σ . Let ˜ Y be the closed 3-manifold obtained from Y ′ by gluing Σ to − Σ and Σ to − Σ using the diffeomorphism h , and let ˜ λ denote thecorresponding multicurve in ˜ Y obtained from λ . Denote by ˜Σ i the image of Σ i in ˜ Y .Kronheimer and Mrowka prove the following excision theorem in [KM10, Theorem 7.7],stated in terms of the notation above; this theorem generalizes work of Floer [Flo90, BD95]in which the surfaces Σ i are assumed to be tori. Theorem 2.12.
There is an isomorphism I ∗ ( Y | Σ ∪ Σ ) λ ∼ = −→ I ∗ ( ˜ Y | ˜Σ ∪ ˜Σ ) ˜ λ , which is homogeneous with respect to the Z / Z -grading, and which, for any submanifold R ⊂ Y disjoint from Σ ∪ Σ , intertwines the actions of µ ( R ) on either side. Remark 2.13.
The fact that the isomorphism in Theorem 2.12 is homogeneous with respectto the Z / Z -grading comes from the fact that it is induced by a cobordism, and such mapsare homogeneous, as in (2.2).The following is a consequence of Theorem 2.12 together with Example 2.9, proved exactlyas in [KM10, Lemma 4.7]. Theorem 2.14.
Let ( Y, λ ) be an admissible pair such that Y fibers over the circle with fibera connected surface R of positive genus with λ · R = 1 . Then I ∗ ( Y | R ) λ ∼ = C . We will prove a converse to this theorem in Section 3, modulo a technical assumption.2.5.
Framed instanton homology.
In this section, we review the construction of framedinstanton homology, which was defined by Kronheimer and Mrowka in [KM11b] and devel-oped further by Scaduto in [Sca15]; it provides a way of assigning an instanton Floer groupto a pair (
Y, λ ) without the admissibility assumption (which requires, e.g., that b ( Y ) > • a basepoint y T in T = T × S , and • a curve λ T = { pt } × S disjoint from this basepoint.Now suppose ( Y, λ ) is a pair consisting of a closed, connected 3-manifold Y and a multicurve λ ⊂ Y . Choose a basepoint y ∈ Y disjoint from λ , and define λ := λ ∪ λ T in Y T , where the connected sum is performed at the basepoints y and y T . Then( Y T , λ )is an admissible pair, which enables us to make the following definition. Definition 2.15.
The framed instanton homology of (
Y, λ ) is the group I ( Y, λ ) := I ∗ ( Y T | T ) λ , where T refers to any T × { pt } ⊂ T disjoint from y T . Remark 2.16.
We will frequently write I ( Y ) for I ( Y, λ ) when [ λ ] = 0 in H ( Y ; Z / Z ),and will generally conflate λ with its mod 2 homology class, given Remark 2.1. Remark 2.17.
Theorem 2.5 implies that the operator µ (pt) : I ∗ ( Y T ) λ → I ∗ ( Y T ) λ satisfies µ (pt) − ≡
0. In particular, the framed instanton homology I ( Y, λ ) ⊂ I ∗ ( Y T ) λ is the honest (i.e. not generalized) 2-eigenspace of µ (pt). Since this operator has degree − Z / Z -grading on I ∗ ( Y T ) λ descends to a relative Z / Z -grading on I ( Y, λ ).As in Remark 2.7, the framed instanton homology I ( Y, λ ) has a canonical Z / Z -grading, I ( Y, λ ) = I ( Y, λ ) ⊕ I ( Y, λ ) , and Scaduto proves the following in [Sca15, Corollary 1.4]. Proposition 2.18.
The Euler characteristic of I ( Y, λ ) is given by χ ( I ( Y, λ )) = ( | H ( Y ; Z ) | b ( Y ) = 00 b ( Y ) > . This motivates the following definition, as in the introduction.
NSTANTONS AND L-SPACE SURGERIES 15
Definition 2.19.
A rational homology 3-sphere Y is an instanton L-space ifdim I ( Y ) = | H ( Y ; Z ) | ;or, equivalently, if I ( Y ) = 0 . A knot K ⊂ S is an instanton L-space knot if S r ( K ) is an instanton L-space for somerational number r > Remark 2.20.
As mentioned in the introduction, we proved in [BS18, Theorem 4.20] thatif the set of positive instanton L-space slopes for a nontrivial knot K is nonempty, then ithas the form [ N, ∞ ) ∩ Q for some positive integer N. Cobordisms induce maps on framed instanton homology as well. Namely, suppose(
X, ν, γ ) : ( Y , λ , y ) → ( Y , λ , y )is a cobordism where γ ⊂ X is an arc from y to y . Following [Sca15, § y and y and a compatible framing of γ , we form a new cobordism X := X ⊲⊳ ( T × [0 , Y T → Y T by removing tubular neighborhoods of γ and y T × [0 ,
1] from X and T × [0 ,
1] respectively,and gluing what remains along the resulting S × [0 , I ( X, ν ) : I ( Y , λ ) → I ( Y , λ )by I ( X, ν )( a ) := I ∗ ( X ) ν ( a ) . Scaduto proves [Sca15, Proposition 7.1] that the grading shift of this map agrees with thatin (2.2), as below.
Proposition 2.21.
The map I ( X, ν ) shifts the Z / Z -grading by deg( I ( X, ν )) = −
32 ( χ ( X ) + σ ( X )) + 12 ( b ( Y ) − b ( Y )) (mod 2) . In particular, this map is homogeneous.
Note that there is a natural inclusion H ( X ; R ) ֒ → H ( X ; R ) . Indeed, any class in H ( X ; R ) can be represented by a multiple of some surface which avoidsthe path γ ; and if such a surface bounds a 3-chain which intersects γ transversely in finitelymany points, then it also bounds a 3-chain in X obtained from the former by replacingthe 3-balls normal to each intersection point with γ with the corresponding punctured T in T × [0 , A ( X ) ֒ → A ( X ) , which enables us to extend the cobordism map I ( X, ν ) to a map D X,ν : I ( Y , λ ) ⊗ A ( X ) → I ( Y , λ )defined by D X,ν ( a ⊗ z ) := Ψ X ,ν ( a ⊗ z ) . In particular, I ( X, ν ) = D X,ν ( − ⊗ . We note below that these cobordism maps automatically satisfy an analogue of the simpletype condition on the Donaldson invariants of closed 4-manifolds. The proof of this lemmais a straightforward application of the relation (2.3) together with the fact that µ (pt) actsas multiplication by 2 on I ( Y , λ ), as in Remark 2.17. Lemma 2.22.
For x = [pt] ∈ H ( X ; R ) and any z ∈ A ( X ) , D X,ν ( − ⊗ x z ) = 4 D X,ν ( − ⊗ z ) as maps from I ( Y , λ ) to I ( Y , λ ) . (cid:3) Remark 2.23.
As indicated above, we will omit the path γ (and the basepoints) from thenotation for these cobordism maps. In practice, we will only consider cobordisms built fromhandle attachments, in which γ is implicitly understood to be a product arc. We may alsoomit ν from the notation for I ( X, ν ) or D X,ν when [ ν ] = 0 in H ( X, ∂X ; Z / Z ), and willgenerally conflate ν with its mod 2 homology class, given Remark 2.11.Given a pair ( Y, λ ) as above and a connected surface R ⊂ Y of positive genus, we will usethe notation I ( Y, λ | R ) to refer to the (2 g ( R ) − µ ( R ) acting on I ( Y, λ ).In particular,(2.5) I ( Y, λ | R ) = I ∗ ( Y T | R ) λ . Moreover, given a cobordism (
X, ν ) : ( Y , λ ) → ( Y , λ )and connected surfaces R ⊂ Y and R ⊂ Y of the same positive genus which are homol-ogous to a surface R ⊂ X , we will denote by I ( X, ν | R ) : I ( Y , λ | R ) → I ( Y , λ | R )the map induced by I ( X, ν ), restricted to the top eigenspaces of µ ( R ) and µ ( R ) (theinduced map respects these eigenspaces, as discussed at the end of § The eigenspace decomposition.
Following [KM10, Corollary 7.6], one can define adecomposition of framed instanton homology which bears some resemblance to the Spin c decompositions of monopole and Heegaard Floer homology, as below. Definition 2.24.
Given a homomorphism s : H ( Y ; Z ) → Z , let I ( Y, λ ; s ) = \ h ∈ H ( Y ; Z ) [ n ≥ ker( µ ( h ) − s ( h )) n be the simultaneous s ( h )-eigenspace of the operators µ ( h ) on I ( Y, λ ), over all h ∈ H ( Y ; Z ). Theorem 2.25.
There is a direct sum decomposition I ( Y, λ ) = M s : H ( Y ; Z ) → Z I ( Y, λ ; s ) . If I ( Y, λ ; s ) is nonzero, then | s ([Σ]) | ≤ g (Σ) − NSTANTONS AND L-SPACE SURGERIES 17 for every connected surface Σ ⊂ Y of positive genus. Thus, only finitely many summandsare nonzero. Finally, there is an isomorphism I ( Y, λ ; s ) ∼ = I ( Y, λ ; − s ) for each s , which preserves the Z / Z -grading.Proof. Remark 2.8 says that for any connected surface Σ ⊂ Y of positive genus, the eigen-values of the operator µ (Σ) acting on I ( Y, λ ) := I ∗ ( Y T | T ) λ are contained in the set of even integers between 2 − g (Σ) and 2 g (Σ) −
2, which immediatelyimplies the second claim of the theorem. The direct sum decomposition then follows fromthe fact that the operators µ ( h ) commute, and have even eigenvalues since each h can berepresented by a connected surface of positive genus. For the last claim, let φ : I ∗ ( Y T ) λ → I ∗ ( Y T ) λ be the map defined in (2.1). Lemma 2.3 implies that for every s : H ( Y ; Z ) → Z , the square φ defines an isomorphism from the ( s ( h ) , µ ( h ) , µ (pt) acting on I ∗ ( Y T ) λ to the ( − s ( h ) , I ( Y, λ ; s ) → I ( Y, λ ; − s ) . It is immediate from the definition of φ that this map preserves the Z / Z -grading. (cid:3) In §
6, we extend the eigenspace decomposition in Theorem 2.25 to cobordism maps.2.7.
The surgery exact triangle.
The following theorem is originally due to Floer [Flo90,BD95], though the formulation below is taken from [Sca15, Theorem 2.1].
Theorem 2.26.
Let Y be a closed, connected 3-manifold, λ ⊂ Y a multicurve, and K ⊂ Y a framed knot with meridian µ ⊂ Y r N ( K ) . Then there is an exact triangle · · · → I ∗ ( Y ) λ → I ∗ ( Y ( K )) λ ∪ µ → I ∗ ( Y ( K )) λ → I ∗ ( Y ) λ → . . . whenever ( Y, λ ) , ( Y ( K ) , λ ∪ µ ) , and ( Y ( K ) , λ ) are all admissible pairs. Moreover, the mapsin this triangle are induced by the corresponding 2-handle cobordisms. This implies the following surgery exact triangle for framed instanton homology, withoutany admissibility hypotheses, as in [Sca15, § Theorem 2.27.
Let Y be a closed, connected 3-manifold, λ ⊂ Y a multicurve, and K ⊂ Y a framed knot with meridian µ ⊂ Y r N ( K ) . Then there is an exact triangle · · · → I ( Y, λ ) → I ( Y ( K ) , λ ∪ µ ) → I ( Y ( K ) , λ ) → I ( Y, λ ) → . . . , in which the maps are induced by the corresponding 2-handle cobordisms. Note that if Y is a homology sphere, and K has framing n relative to its Seifert framing,then by taking λ = ( n odd µ n even , we ensure that the curves λ and λ ∪ µ appearing in Theorem 2.27 (and Theorem 2.26) arezero in homology over Z / Z . In the case of Y = S , for instance, this choice of λ thereforeyields an exact triangle(2.6) · · · → I ( S ) → I ( S n ( K )) → I ( S n +1 ( K )) → . . . . The connected sum theorem.
We will make use of the following version of Fukaya’sconnected sum theorem [Fuk96], as applied by Scaduto in [Sca15].
Theorem 2.28.
Let ( Y, λ ) be an admissible pair. Then there is an isomorphism of relatively Z / Z -graded C -modules, I ( Y, λ ) ∼ = ker( µ (pt) − ⊗ H ∗ ( S ; C ) , where µ (pt) − above is viewed as acting on four consecutive gradings of I ∗ ( Y ) λ . In light of Theorem 2.5, this implies the following, as in [Sca15, § Corollary 2.29.
Let ( Y, λ ) be an admissible pair such that λ intersects a surface of genusat most 2 in an odd number of points. Then I ( Y, λ ) ⊗ H ∗ ( S ; C ) ∼ = I ∗ ( Y ) λ ⊗ H ∗ ( S ; C ) as relatively Z / Z -graded C -modules. Instanton homology and fibered manifolds
In this section, we prove a converse to Theorem 2.14—namely, that if I ∗ ( Y | R ) λ ∼ = C , then Y is fibered with fiber R —modulo a technical assumption which suffices for our applications.We will use this converse, Theorem 3.2, to show that instanton Floer homology of 0-surgeryon a knot detects whether the knot is fibered, as in Theorem 3.7 below, and likewise forframed instanton homology, as in Theorem 1.17, proved in §
4. The latter will then be usedin our proof that instanton L-space knots are fibered, as outlined in the introduction.The statement of Theorem 3.2 requires the following definition.
Definition 3.1.
A compact 3-manifold M with boundary Σ + ⊔ Σ − is a homology product if both of the maps ( i ± ) ∗ : H ∗ (Σ ± ; Z ) → H ∗ ( M ; Z )induced by inclusion are isomorphisms. Theorem 3.2.
Let ( Y, λ ) be an admissible pair with Y irreducible, and R ⊂ Y a connectedsurface of positive genus with λ · R = 1 . If Y r N ( R ) is a homology product and I ∗ ( Y | R ) λ ∼ = C , then Y is fibered over the circle with fiber R . Remark 3.3.
We would like to prove Theorem 3.2 without the homology product assump-tion. Indeed, the corresponding results in monopole and Heegaard Floer homology arestated without it because one can show in those cases that rank 1 automatically implies thisassumption. The key input there is the fact that the Turaev torsion of Y is generally equalto the Euler characteristic of monopole and Heegaard Floer homology, but no such resultsare known for instanton homology. NSTANTONS AND L-SPACE SURGERIES 19
We will reduce Theorem 3.2 to the special case in Proposition 3.5 by a standard argument,exactly as in [Ni07, Ni09b, KM10, Ni08]. Before stating the proposition, another definition.
Definition 3.4.
A homology product M with boundary Σ + ⊔ Σ − is vertically prime if everyclosed, connected surface in M in the same homology class and with the same genus as Σ + is isotopic to either Σ + or Σ − . Proposition 3.5.
Theorem 3.2 holds under the additional assumption that M = Y r N ( R ) is vertically prime.Proof. Our proof is a straightforward combination of the proof of [KM10, Theorem 7.18],which asserts that sutured instanton homology detects products, with the work in [Ni08],which uses the analogous result for sutured monopole homology plus arguments from [Ni07,Ni09a] to conclude that monopole Floer homology detects fibered 3-manifolds. In particular,we follow Ni’s argument and terminology from [Ni08] nearly to the letter.Let
E ⊂ H ( M ) be the subgroup spanned by the homology classes of product annuli in M , whose boundary we write as R + ⊔ R − . Following Ni, we will first show that E = H ( M ).Suppose that E 6 = H ( M ). Then there exist essential simple closed curves ω − ⊂ R − and ω + ⊂ R + which are homologous in M and satisfy [ ω ± ]
6∈ E . In this case, Ni fixes an arc σ from R − to R + , and constructs for any sufficiently large m connected surfaces S , S ⊂ M with ∂S = ω − − ω + and σ · S = m,∂S = ω + − ω − and σ · S = m, such that decomposing M (viewed as a sutured manifold with an empty suture) along either S or S produces a taut sutured manifold [Ni08, Ni07]. Now choose a diffeomorphism h : R + → R − such that Z = M/h is a homology R × S , h ( ω + ) = ω − , and h ( R + ∩ λ ) = h ( R − ∩ λ ) , so that λ extends to a closed curve ¯ λ ⊂ Z . As in the proof of [KM10, Theorem 7.18], wecan arrange that the closed surfaces ¯ R = R + /h , ¯ S = S /h , and ¯ S = S /h in Z satisfy[ ¯ S ] = m [ ¯ R ] + [ ¯ S ] , (3.1) [ ¯ S ] = m [ ¯ R ] − [ ¯ S ] , (3.2) χ ( ¯ S ) = χ ( ¯ S ) = mχ ( ¯ R ) + χ ( ¯ S ) , (3.3)for some closed surface ¯ S ⊂ Z with ¯ λ · ¯ S = 0 and2 g ( ¯ S ) − > S = S /h for some connected surface S ⊂ M which provides a homology between ω + and ω − and satisfies σ · S = 0; S is not an annulussince [ ω ± ]
6∈ E ).Following the proof of [KM10, Theorem 7.18], we let T i ⊂ Z be the closed, connectedsurface obtained by smoothing out the circle of intersection ¯ R ∩ ¯ S i for i = 1 ,
2. Then2 g ( T i ) − g ( ¯ R ) −
2) + (2 g ( ¯ S i ) − . Moreover, T i is genus-minimizing in its homology class since decomposing Z along T i yieldsthe same taut sutured manifold as decomposing M along S i (a decomposing surface must begenus-minimizing if the resulting decomposition is taut). Let us suppose m is even so that¯ λ · T i is odd. Then Theorem 2.10 implies that I ∗ ( Z | T i ) ¯ λ = 0 for each i . Let x i be a nonzeroelement of I ∗ ( Z | T i ) ¯ λ for i = 1 ,
2. Since this element lies in the (2 g ( T i ) − µ ( T i ) = µ ( ¯ R ) + µ ( ¯ S i ) , and the eigenvalues of µ ( ¯ R ) and µ ( ¯ S i ) are at most 2 g ( ¯ R ) − g ( ¯ S i ) −
2, it follows that x i lies in these top eigenspaces of µ ( ¯ R ) and µ ( ¯ S i ) as well. Thus, we have nonzero elements x , x ∈ I ∗ ( Z | ¯ R ) ¯ λ such that x i lies in the (2 g ( ¯ S i ) − µ ( ¯ S i ). It then follows from the identities(3.1)-(3.3) that x and x also lie in the top eigenspaces of µ ( ¯ S ) and − µ ( ¯ S ), respectively.But these eigenspaces are disjoint since 2 g ( ¯ S ) − >
0. Therefore,dim I ∗ ( Z | ¯ R ) ¯ λ ≥ . On the other hand, we have I ∗ ( Z | ¯ R ) ¯ λ ∼ = I ∗ ( Y | R ) λ ∼ = C by Theorem 2.12, a contradiction. We conclude that E = H ( M ) after all.In the terminology of [Ni08, Corollary 6], the fact that E = H ( M ) implies that the map i ∗ : H (Π) → H ( M )is surjective, where (Π , Ψ) is the characteristic product pair for (
M, ∂M ). According to Ni,we can therefore find an embedded G × [ − ,
1] inside M , where G ⊂ R + is a genus one surfacewith boundary obtained as the tubular neighborhood of two curves in R + intersecting in asingle point. Let M ′ = M r (int( G ) × [ − , γ ′ = ∂G × { } ⊂ ∂M ′ . Then ( M ′ , γ ′ ) is a sutured manifold and a homology product. By definition, its suturedinstanton homology is given by SHI ( M ′ , γ ′ ) := I ∗ ( Y | R ) λ ∼ = C . ( M ′ , γ ′ ) is taut since this module is nonzero, so [KM10, Theorem 7.18] asserts that ( M ′ , γ ′ )is a product sutured manifold. In particular, M ∼ = R × [ − , , which implies that Y is fibered with fiber R . (cid:3) As in [Ni08], Theorem 3.2 follows easily from Proposition 3.5 and excision.
Proof of Theorem 3.2.
Suppose the hypotheses of the theorem are satisfied. Take a maximalcollection R = R , R , . . . , R n − of disjoint, pairwise non-isotopic closed surfaces in Y whichhave the same genus as R and are homologous to R . Let us write Y r ( N ( R ) ⊔ N ( R ) ⊔ · · · ⊔ N ( R n − )) = M ⊔ M ⊔ · · · ⊔ M n − , in which we have ordered the R i so that ∂M i = R i ⊔ R i +1 , interpreting the subscripts mod n .For each i , form a closed manifold Y i and curve λ i ⊂ Y i from M i by taking a diffeomorphism NSTANTONS AND L-SPACE SURGERIES 21 R i → R i +1 which identifies λ ∩ R i with λ ∩ R i +1 , and using this map to glue R i to R i +1 .By repeated application of Theorem 2.12, we have that I ∗ ( Y | R ) λ ∼ = I ∗ ( Y | R ) λ ⊗ · · · ⊗ I ∗ ( Y n − | R n − ) λ n − . Then I ∗ ( Y | R ) λ ∼ = C implies that I ∗ ( Y i | R i ) λ i ∼ = C for all i . Moreover, the fact that Y r N ( R )is a homology product implies that each M i ∼ = Y i r N ( R i )is as well, by [Ni09b, Lemma 4.2]. Finally, each M i is vertically prime since the collection { R i } is maximal, so Proposition 3.5 says that M i ∼ = R i × [ − ,
1] for all i . It follows that Y is fibered over S with fiber R , as desired. (cid:3) The remainder of this section is devoted to proving Theorem 3.7.Suppose K is a nontrivial knot in S , and let ˆΣ ⊂ S ( K ) be the closed surface of genus g = g ( K ) formed by capping off a genus- g Seifert surface Σ for K with a disk. Let µ ⊂ S ( K )be the image of the meridian of K in the surgered manifold, with µ · ˆΣ = 1. For each integer j = 1 − g, − g, . . . , g − , let I ∗ ( S ( K ) , ˆΣ , j ) µ ⊂ I ∗ ( S ( K )) µ denote the (2 j, µ ( ˆΣ) , µ (pt) (these eigenspaces are trivial for | j | ≥ g by Theorem 2.4). Since the Z / Z -grading on I ∗ ( S ( K )) µ is fixed by µ ( ˆΣ) and µ (pt),each I ∗ ( S ( K ) , ˆΣ , j ) µ inherits this grading and hence has a well-defined Euler characteristic.Lim proved the following [Lim10, Corollary 1.2], which says that these Euler characteristicsare determined by the Alexander polynomial of K . Theorem 3.6.
Let ∆ K ( t ) be the Alexander polynomial of K , normalized so that ∆ K ( t ) = ∆ K ( t − ) and ∆ K (1) = 1 . Then ∆ K ( t ) − t − t − = g − X j =1 − g χ ( I ∗ ( S ( K ) , ˆΣ , j ) µ ) t j . We will use Lim’s result in combination with Theorem 3.2 to prove the following fibered-ness detection theorem, stated in terms of the notation above.
Theorem 3.7.
The C -module I ∗ ( S ( K ) | ˆΣ) µ is always nontrivial, and I ∗ ( S ( K ) | ˆΣ) µ ∼ = C iff K is fibered.Proof. Gabai [Gab87] proved that S ( K ) is irreducible and that ˆΣ minimizes genus withinits homology class iff its genus is equal to the Seifert genus g = g ( K ), so the nontriviality isa consequence of Theorem 2.10. He also proved in [Gab87] that K is fibered with fiber Σ iff S ( K ) is fibered with fiber ˆΣ. We already know that if S ( K ) is fibered then I ∗ ( S ( K ) | ˆΣ) µ ∼ = C , by Theorem 2.14. Let us prove the converse. Suppose I ∗ ( S ( K ) | ˆΣ) µ ∼ = C . We claim that S ( K ) r N ( ˆΣ) is a homology product. Indeed,the Euler characteristic χ ( I ∗ ( S ( K ) , ˆΣ , g − µ ) = χ ( I ∗ ( S ( K ) | ˆΣ) µ )is ±
1, so Theorem 3.6 says that∆ K ( t ) = 1 + ( t − t − ) · ( ± t g − + . . . ) = ± t g + . . . , where the omitted terms have strictly lower degree. The Alexander polynomial ∆ K ( t ) istherefore monic of degree g . This implies that S ( K ) r N ( ˆΣ) is a homology product by[Ghi08, Lemma 4.10]. We may then conclude from Theorem 3.2 that S ( K ) is fibered withfiber ˆΣ, and hence that K is fibered with fiber Σ. (cid:3) Framed instanton homology and fibered knots
The main goal of this section is to prove Theorem 1.17, which asserts that framed instan-ton homology of 0-surgery on a nontrivial knot detects whether the knot is fibered. Notethat Theorem 1.17 follows immediately from Theorem 3.7 if we can prove that the relevant C -modules in the two theorems have the same dimensions,(4.1) dim I ( S ( K ) , µ ; s g − ) = dim I ∗ ( S ( K ) | ˆΣ) µ , where g = g ( K ) = g ( ˆΣ). Recall from § i ∈ Z , s i : H ( S ( K )) → Z is the homomorphism defined by s i ([ ˆΣ]) = 2 i. So, from § I ( S ( K ) , µ ; s g − )is, by definition, the (2 g − µ ( ˆΣ) acting on(4.2) I ( S ( K ) , µ ) := I ∗ ( S ( K ) T | T ) µ . Since the module in (4.2) is precisely the 2-eigenspace of µ (pt) acting on I ∗ ( S ( K ) T ) µ , per Remark 2.17, the top eigenspace of µ ( ˆΣ) acting on the module in (4.2) is the same as I ∗ ( S ( K ) T | ˆΣ) µ , by Definition 2.6. In conclusion, we have that(4.3) I ( S ( K ) , µ ; s g − ) = I ∗ ( S ( K ) T | ˆΣ) µ . The equality in (4.1), which proves Theorem 1.17, will then follow if we can show that thereis an isomorphism(4.4) I ∗ ( S ( K ) T | ˆΣ) µ ∼ = I ∗ ( S ( K ) | ˆΣ) µ ⊗ ( C ⊕ C )which is homogeneous with respect to the Z / Z -grading, where C i is a copy of C in grading i ; indeed, (4.3) and (4.4) will imply thatdim I ( S ( K ) , µ ; s g − ) = dim I ( S ( K ) , µ ; s g − ) = dim I ∗ ( S ( K ) | ˆΣ) µ . Our focus below will therefore be on proving the isomorphism in (4.4), though we will workin greater generality for most of this section.
NSTANTONS AND L-SPACE SURGERIES 23
Indeed, rather than considering ( S ( K ) , µ ) and ˆΣ, we let ( Y, λ ) be an arbitrary admissiblepair, and R ⊂ Y a connnected surface of positive genus with λ · R odd. We note exactly asabove that(4.5) I ∗ ( Y T | T ) λ is the 2-eigenspace of µ (pt) acting on I ∗ ( Y T ) λ , which means that the top eigenspace of µ ( R ) acting on the module in (4.5) is simply I ∗ ( Y T | R ) λ . The first of our two main propositions is the following.
Proposition 4.1.
Suppose ( Y, λ ) is an admissible pair, and R ⊂ Y is a connected surfaceof positive genus with λ · R = 1 . Then there is an isomorphism I ∗ ( Y T | R ) λ ∼ = I ∗ ( Y S × S ) | R ) λ , which is homogeneous with respect to the Z / Z -grading, where the connected sum with S × S is performed away from R and λ .Proof. Recall from the definition of framed instanton homology in § λ = λ ∪ λ T , where λ T = { pt } × S is a curve dual to T = T × { pt } in T = T × S . We can interpret I ∗ ( Y T | T ) λ ∪ λ T as a version of the sutured instanton homology SHI ( Y r B , S ) defined in [KM10], albeitone in which Y r B is equipped with the nontrivial bundle specified by λ . Fixing a surface R ′ ∼ = R , it thus follows exactly as in the proof of invariance of SHI in [KM10, § I ∗ ( Y T | T ) λ ∪ λ T ∼ = I ∗ ( Y R ′ × S ) | R ′ ) λ ∪ λ R ′ , where λ R ′ = { pt } × S is a curve dual to R ′ = R ′ × { pt } in R ′ × S . The isomorphism (4.6)is obtained as a composition of excision isomorphisms along tori that are disjoint from R . Ittherefore intertwines the actions of µ ( R ) on either side, as in Theorem 2.12. In particular,this isomorphism identifies the 2 j -eigenspace of µ ( R ) acting on I ∗ ( Y T | T ) λ ∪ λ T with the2 j -eigenspace of µ ( R ) acting on I ∗ ( Y R ′ × S ) | R ′ ) λ ∪ λ R ′ . In the case j = g ( R ) −
1, this isomorphism becomes(4.7) I ∗ ( Y T | R ) λ ∪ λ T ∼ = I ∗ ( Y R ′ × S ) | R ∪ R ′ ) λ ∪ λ R ′ . This isomorphism is homogeneous with respect to the Z / Z -grading, as in Theorem 2.12.To prove the proposition, we now apply excision once more. Namely, we cut Y R ′ × S )open along R and R ′ × { pt } , and then glue the two resulting R components of the boundaryto the two R ′ components via some identification R ∼ = R ′ , as illustrated in Figure 1. We canassume this identification is such that the multicurve λ ∪ λ T is cut and reglued to form λ inthe resulting manifold, which is Y S × S ). By Theorem 2.12, there is an isomorphism(4.8) I ∗ ( Y R ′ × S ) | R ∪ R ′ ) λ ∪ λ R ′ ∼ = I ∗ ( Y S × S ) | ˜ R ∪ ˜ R ) λ , which is homogeneous with respect to the Z / Z -grading, in which ˜ R and ˜ R are the imagesof R and R ′ . We may identify one of these two surfaces—say ˜ R —with R ⊂ Y S × S ) . S × [0 , → RR ′ YR ′ × S ❀ Y r N ( R ) R ′ × [0 , ❀ ˜ R ˜ R Figure 1.
Relating Y R ′ × S ) to Y S × S ) by excision. The thincurves are meant to represent λ ∪ λ R ′ on the left, and λ on the right.Furthermore, note that ˜ R and ˜ R cobound a region in Y S × S ) with one of the 2-spheres in the connected sum neck. Adding a trivial handle to this 2-sphere, we obtain atorus T in Y S × S ) such that [ ˜ R ] − [ ˜ R ] = [ T ]in homology. Since µ ( T ) is nilpotent, a generalized (2 g ( R ) − R is also ageneralized (2 g ( R ) − R , and vice versa. We therefore have that(4.9) I ∗ ( Y S × S ) | ˜ R ∪ ˜ R ) λ ∼ = I ∗ ( Y S × S ) | ˜ R ) λ = I ∗ ( Y S × S ) | R ) λ . Combining (4.7), (4.8), and (4.9) then completes the proof of the proposition. (cid:3)
Our second main proposition of this section is the following.
Proposition 4.2.
Suppose ( Y, λ ) is an admissible pair, and R ⊂ Y is a connected surfaceof positive genus with λ · R odd. Then there is an isomorphism of Z / Z -graded C -modules I ∗ ( Y S × S ) | R ) λ ∼ = I ∗ ( Y | R ) λ ⊗ ( C ⊕ C ) , where each C i is a copy of C in grading i .Proof. We model our argument after the computation of I ( S × S ) in [Sca15, § U denote an unknot in Y . We apply Floer’s surgery exact triangle, Theorem 2.26, for surgerieson U with framings ∞ , 0, and 1 to get an exact triangle · · · → I ∗ ( Y ) λ F −→ I ∗ ( Y S × S )) λ G −→ I ∗ ( Y ) λ H −→ . . . (where we take the λ in that theorem to be the union of our λ with the µ in the theorem).The maps F , G , and H are induced by 2-handle cobordisms where the 2-handles are attachedaway from R , so these maps intertwine the actions of µ ( R ) on these modules. The exacttriangle above therefore restricts to an exact triangle(4.10) · · · → I ∗ ( Y | R ) λ F −→ I ∗ ( Y S × S ) | R ) λ G −→ I ∗ ( Y | R ) λ H −→ . . . Since F has odd degree while G has even degree, by the formula (2.2), the proposition willfollow if we can show that H = 0.The map H is induced by a cobordism( X, ν ) : (
Y, λ ) → ( Y, λ ) NSTANTONS AND L-SPACE SURGERIES 25 built by attaching a ( − Y × [0 ,
1] along the unknot U × { } . In otherwords, X ∼ = ( Y × [0 , CP . Letting e ∈ CP denote the exceptional sphere, the pairing ν · e was determined to be odd in[Sca15, § H must be zero by a dimension-countingargument, exactly as in [BD95, § S , we can realize aninstanton A on X by gluing instantons A Y × [0 , and A CP on either summand. The formerare irreducible and the latter have stabilizer at most S , since the bundle specified by ν isnontrivial on CP ; and the unique flat connection on S has 3-dimensional stabilizer, so wehave ind( A ) = ind( A Y × [0 , ) + ind( A CP ) + 3 ≥ −
1) + 3 = 2 . Thus, there are generically no index-0 instantons on X , which implies that H is zero. (cid:3) Combining Propositions 4.1 and 4.2, we have the following immediate corollary.
Corollary 4.3.
Suppose ( Y, λ ) is an admissible pair, and R ⊂ Y is a connected surface ofpositive genus with λ · R = 1 . Then there is an isomorphism of C -modules, I ∗ ( Y T | R ) λ ∼ = I ∗ ( Y | R ) λ ⊗ ( C ⊕ C ) , which is homogeneous with respect to the Z / Z -grading.Proof of Theorem 1.17. Applying Corollary 4.3 to the case in which Y = S ( K ), R = ˆΣ,and λ = µ , we obtain the isomorphism (4.4) from which Theorem 1.17 follows, as discussedat the beginning of this section. (cid:3) With respect to the notation I ( Y, λ | R ) := I ∗ ( Y T | R ) λ of (2.5), Corollary 4.3 together with Theorem 2.14 immediately imply the following. Proposition 4.4.
Let ( Y, λ ) be an admissible pair such that Y fibers over the circle withfiber a connected surface R of positive genus with λ · R = 1 . Then I ( Y, λ | R ) ∼ = I ( Y, λ | R ) ∼ = C . A structure theorem for cobordism maps
Following [KM95a], we let x = [pt] in H ( X ; Z ) and then define a formal power series bythe formula D νX ( h ) = D X,ν (cid:16) − ⊗ (cid:16) e h + xe h (cid:17)(cid:17) for each h ∈ H ( X ; R ). Our goal over the next several subsections is to prove the followingstructure theorem for the formal cobordism maps D νX . Theorem 5.1.
Let ( X, ν ) : ( Y , λ ) → ( Y , λ ) be a cobordism with b ( X ) = 0 . Then thereis a finite collection (possibly empty) of basic classes K , . . . , K r : H ( X ; Z ) → Z , each satisfying K j ( α ) ≡ α · α (mod 2) for all α ∈ H ( X ; Z ) ; and nonzero elements a , . . . , a r ∈ Hom( I ( Y , λ ) , I ( Y , λ )) with rational coefficients , such that for any α ∈ H ( X ; Z ) we have D ν + αX ( h ) = e Q ( h ) / r X j =1 ( − ( K j ( α )+ α · α )+ ν · α · a j e K j ( h ) . The basic classes satisfy an adjunction inequality | K j ([Σ]) | + [Σ] · [Σ] ≤ g (Σ) − for all smoothly embedded surfaces Σ ⊂ X of genus g (Σ) ≥ and positive self-intersection. Theorem 5.1 is a direct analogue of Kronheimer and Mrowka’s structure theorem for theDonaldson invariants of closed 4-manifolds of simple type [KM95a], which was subsequentlygiven different proofs by Fintushel and Stern [FS95] and Mu˜noz [Mu˜n00]. None of theseproofs applies verbatim to an arbitrary cobordism: the proofs in [KM95a, Mu˜n00] requirea smoothly embedded surface of positive self-intersection, while the one in [FS95] requires π ( X ) = 1 and involves interaction between both SO (3) and SU (2) invariants. We will proveTheorem 5.1 by adapting Mu˜noz’s proof, which is short and only requires two ingredients:a blow-up formula and a computation of Fukaya–Floer homology for products Σ × S .Our proof of Theorem 5.1 is structured as follows. We first verify in § § b +2 ( X ) >
0. In § T , induces an injectivemap on I , and we use this in § b +2 ( X ) is positive. Remark 5.2.
Some of the results which we adapt from the closed 4-manifold setting (likeFintushel–Stern’s blow-up formula) require that the closed manifold have b +2 > b +2 > X limits at theends Y i T to a flat connection on an admissible bundle, so it is automatically irreducibleat the ends and hence on X itself. Furthermore, these maps are well-defined without anyassumption on b +2 .Some of these results also require the simple type assumption, but this is automaticallysatisfied for the cobordisms X by Lemma 2.22.5.1. The blow-up formula for cobordism maps.
The blow-up formula for formal cobor-dism maps reads as follows.
Theorem 5.3.
Let ( X, ν ) : ( Y , λ ) → ( Y , λ ) be a cobordism, and let ˜ X = X CP denotethe blow-up of X at a point. Let E denote the Poincar´e dual of the class e ∈ H ( ˜ X ; Z ) ofan exceptional sphere. Then D ν ˜ X = D νX · e − E / cosh E D ν + e ˜ X = − D νX · e − E / sinh E Ultimately, framed instanton homology can be defined over Z ; what we mean here is that the a j haverational coefficients with respect to rational bases for I ( Y , λ ) and I ( Y , λ ). We remark that the resultsof this section hold over any field of characteristic zero. NSTANTONS AND L-SPACE SURGERIES 27 as formal
Hom( I ( Y , λ ) , I ( Y , λ )) -valued functions on H ( X ; R ) . This was proven by Fintushel and Stern for the Donaldson invariants of closed 4-manifoldsin [FS96]. We explain briefly why their argument carries over to the case of cobordisms.
Proof of Theorem 5.3.
The blow-up formula of [FS96] is a formal consequence of a handful ofconcrete identities given in [FS96, § D X,ν in the presenceof a sphere of self-intersection − −
3, and which relate D X,ν to D ˜ X,ν and D ˜ X,ν + e . Theseidentities were originally proven in the setting of closed 4-manifolds but are local, obtainedvia an analysis of ASD connections in neighborhoods of such spheres, and therefore hold forcobordism maps, so the blow-up formula remains valid in this setting. The formulas herematch the special case [FS96, Theorem 5.2], whose proof requires only the assumption thatthe closed 4-manifold have simple type, which is automatically satisfied by the cobordisms X as noted in Remark 5.2 and proved in Lemma 2.22. (cid:3) The structure theorem for b +2 positive. In this subsection we prove the following.
Proposition 5.4.
Theorem 5.1 holds for cobordisms X with b +2 ( X ) > . Our proof of Proposition 5.4 follows the proof of [Mu˜n00, Theorem 1.2], given in [Mu˜n00, § X with strong simple type , which means that for any Hermitian line bundle w → X and z ∈ A ( X ), the Donaldson invariants satisfy D X,w (( x − z ) = 0 and D X,w ( γz ) = 0for all γ ∈ H ( X ). In our situation, the first of these is Lemma 2.22, and we ignore thesecond by requiring that b ( X ) = 0.The key input in Mu˜noz’s proof is the following lemma [Mu˜n00, Lemma 3.1]. Lemma 5.5.
Let ( X, ν ) : ( Y , λ ) → ( Y , λ ) be a cobordism with b ( X ) = 0 . Let Σ ⊂ X bea surface of genus g ≥ , with Σ · Σ = 0 and ν · Σ odd. Then for any other class D ∈ H ( X ) there are power series f r,D ( t ) , with − g ≤ r ≤ g − , such that D νX ( tD + s Σ) = e Q ( tD + s Σ) / g − X r =1 − g f r,D ( t ) e rs D ν +Σ X ( tD + s Σ) = e Q ( tD + s Σ) / g − X r =1 − g ( − r +1 f r,D ( t ) e rs . Remark 5.6.
Mu˜noz states this lemma for closed 4-manifolds in [Mu˜n00, Lemma 3.1], butit applies verbatim to the cobordisms maps here. Indeed, the proof only uses the relationsimposed by the relative invariants of a Σ × D neighborhood of Σ ⊂ X in the Fukaya–Floerhomology HFF (Σ × S , S ). In particular, Mu˜noz uses his computation of this Fukaya–Floerhomology from [Mu˜n99a] to deduce that both series are annihilated by the operator g − Y r =1 − g (cid:18) ∂∂s − (2 r + t ( D · Σ)) (cid:19) , from which the lemma follows.We now prove Proposition 5.4 via a series of lemmas, corresponding to steps 2 through5 of [Mu˜n00, § which is automatically satisfied in our case.) Each of these is labeled with the correspondingstep from [Mu˜n00], and we indicate explicitly how our proofs differ from those in [Mu˜n00].Throughout this subsection we assume that( X, ν ) : ( Y , λ ) → ( Y , λ )is a cobordism with b ( X ) = 0 and b +2 ( X ) > Lemma 5.7 (Step 2) . For any cobordism ν : λ → λ in X , there are finitely many nonzero a ν,i ∈ Hom( I ( Y , λ ) , I ( Y , λ )) and homomorphisms K ν,i : H ( X ; Z ) → Z such that D νX ( h ) = e Q ( h ) / X i a ν,i e K ν,i ( h ) for all h ∈ H ( X ; Z ) . As originally stated, [Mu˜n00, Step 2] only proves the corresponding property for a single ν , but we will establish it for all ν simultaneously. Proof of Lemma 5.7.
Just as in [Mu˜n00], it suffices to prove this for some blow-up of X and some ˜ ν which is homologous to a sum of ν and some exceptional spheres, since if˜ X = X CP has exceptional divisor e then we have D νX ( h ) = D ν ˜ X ( h ) and D νX ( h ) = − ddr D ν + e ˜ X ( h + re ) (cid:12)(cid:12)(cid:12) r =0 by the blow-up formula of Theorem 5.3.We begin by finding a convenient basis of H ( X ); the construction in [Mu˜n00] uses thefact that in the closed case, after blowing up we can arrange Q X = a (1) ⊕ b ( − X a cobordism. Instead, using the assumption that b +2 ( X ) >
0, we letΣ , . . . , Σ k be any integral basis of H ( X ) / torsion with Σ · Σ >
0. For j ≥
2, we replaceeach Σ j with Σ j + n Σ for a suitably large n , and then we still have an integral basis butnow Σ j · Σ j > j ≥ n i = Σ i · Σ i for 1 ≤ i ≤ k and let ˜ X be the blow-up of X at P ki =1 n i points, with e i,j for 1 ≤ i ≤ k, ≤ j ≤ n i being the exceptional spheres. For 1 ≤ i ≤ k we define a collection of homology classes by S i = Σ i − e i, − e i, − · · · − e i,n i S ji = Σ i − e i, − · · · + e i,j − · · · − e i,n i , ≤ j ≤ n i . These are integral classes which span H ( ˜ X ; Q ). In fact, if we let H = span Z ( { S ji | ≤ i ≤ k, ≤ j ≤ n i } ) ⊂ H ( ˜ X ; Z ) / torsion , then from the relations 2 e i,j = S ji − S i for all j ≥
1, and2Σ i = 2 S i + 2 n i X j =1 e i,j = (2 − n i ) S i + 2 n i X j =1 S ji , we deduce that 2 H ( ˜ X ; Z ) / torsion ⊂ H. NSTANTONS AND L-SPACE SURGERIES 29
We also define a class ˜ ν ∈ H ( ˜ X, ∂ ˜ X ) by˜ ν = ν + k X i =1 c i e i, , where c i = ( ν · Σ i odd1 ν · Σ i even , so that for all i and j we have ˜ ν · S ji ≡ S ji is represented by a smoothly embedded surface of genus g i,j ≥
1, thenjust as in [Mu˜n00], repeated application of Lemma 5.5 says that(5.1) D ˜ ν ˜ X k X i =1 n i X j =0 t i,j S ji = e Q ( P t i,j S ji ) / X − g i,j ≤ r i,j ≤ g i,j − a { r i,j } e P r i,j t i,j for some a { r i,j } : I ( Y , λ ) → I ( Y , λ ) . The exponents P r i,j t i,j are valued in 2 Z on all of H and in particular on 2 H ( ˜ X ; Z ) / torsion,so they define homomorphisms H ( ˜ X ; Z ) → Z . Moreover, the a { r i,j } can in fact be taken tohave rational coefficients, because the same is true for each D ˜ X, ˜ ν ( z ) where z has the form h d or x h d with h ∈ H ( ˜ X ; Z ). (cid:3) Lemma 5.8 (Step 3) . The basic classes K ν,ℓ of Lemma 5.7 satisfy K ν,ℓ ( h ) + h · h ≡ for all h ∈ H ( X ; Z ) . Our argument here differs from the one in [Mu˜n00], because that one asserts that for any x ∈ H ( ˜ X ) there are some i and j such that x · S ji = 0, and this need not be true when X is a cobordism rather than a closed 4-manifold. Proof of Lemma 5.8.
Both terms are linear mod 2 in h , so it suffices to prove the claim forthe basis { Σ , . . . , Σ k } of H ( X ; Z ) / torsion which we used in the proof of Lemma 5.7. Weborrow notation from that proof, blowing up X to get ˜ X and surfaces S ji exactly as before.We observe for each i that if K is any basic class of ( ˜ X, ˜ ν ), then equation (5.1) says that K ( S i ) is even. Letting E i,j denote the Poincar´e duals of the exceptional spheres e i,j in ˜ X ,the blow-up formula (Theorem 5.3) says that K = K ν,ℓ + k X i =1 n i X j =1 E i,j is a basic class of ( ˜ X, ˜ ν ). Since S i = Σ i − P n i j =1 e i,j for all i , we have K ( S i ) = K ν,ℓ (Σ i ) + n i = K ν,ℓ (Σ i ) + Σ i · Σ i and hence this last expression is even for all i , as desired. (cid:3) Lemma 5.9 (Step 4) . Fix ν and let a ν,i and K ν,i be the coefficients and basic classesappearing in Lemma 5.7. For any class α ∈ H ( X ; Z ) , we have D ν + αX ( h ) = e Q ( h ) / X i ( − ( K ν,i ( α )+ α · α )+ ν · α · a ν,i e K ν,i ( h ) . In other words, the basic classes K i = K ν,i do not depend on the particular choice of ν , andtheir coefficients are related by a ν + α,i = ( − ( K ν,i ( α )+ α · α )+ ν · α · a ν,i for all i . Our argument here is nearly the same as in [Mu˜n00], but with a different choice of signsin the definition of Σ which avoids the need for the identity D X,ν +2 α = ( − α D X,ν . Proof of Lemma 5.9.
We note that the exponent12 ( K ν,i ( α ) + α · α ) + ν · α is integral by Lemma 5.8, and its reduction modulo 2 is linear in α . Since H ( X ; Z ) / torsionhas an integral basis consisting of surfaces of positive self-intersection, as seen in the proofof Lemma 5.7, it therefore suffices to prove the proposition when N = α · α is positive. Wewill let ˜ X be the N -fold blow-up of X , with exceptional spheres e , . . . , e N and E j = PD ( e j )as usual.Suppose first that ν · α is odd. Letting Σ = α + P j e j , so that ν · Σ is odd and Σ · Σ = 0,we have D ν ˜ X = D νX · e − P j E j / N Y j =1 cosh E j D ν +Σ˜ X = D ν + αX · e − P j E j / N Y j =1 ( − sinh E j )by the blow-up formula. Lemma 5.5 tells us that ( ˜ X, ν ) and ( ˜
X, ν + Σ) have the same basicclasses, and these are K ν,i + N X j =1 σ j E j and K ν + α,i + N X j =1 σ j E j , σ j ∈ {± } respectively, so ( X, ν ) and (
X, ν + α ) must have the same basic classes as well, say K ν,i = K ν + α,i . Moreover, the lemma says that the coefficient of K = K ν,i + P j E j in D ν +Σ˜ X is a ν,i N · ( − K (Σ)+1 = a ν,i N · ( − ( K ν,i ( α ) − N )+1 , while by the blow-up formula it is ( − N · a ν + α,i N . Equating the two, and recalling that N = α · α and ν · α ≡ a ν + α,i = α ν,i · ( − ( K ν,i ( α )+ α · α )+ ν · α as claimed. NSTANTONS AND L-SPACE SURGERIES 31
Now suppose instead that ν · α is even. This time we take Σ = α − e + P Nj =2 e j , so thatΣ · Σ = 0 and ( ν + e ) · Σ is odd. Then the blow-up formula gives D ν + e ˜ X = D νX · e − P j E j / ( − sinh E ) N Y j =2 cosh E j D ν + e +Σ˜ X = D ν + αX · e − P j E j / (cosh E ) N Y j =2 ( − sinh E j ) . Applying Lemma 5.5 again, we conclude exactly as before that (
X, ν ) and (
X, ν + α ) havethe same basic classes K ν,i = K ν + α,i ; and since K = K ν,i + P j E j has coefficient − a ν,i / N in D ν + e ˜ X , the lemma also says that its coefficient in D ν + e +Σ˜ X is − a ν,i N · ( − K (Σ)+1 = a ν,i N · ( − ( K ν,i ( α ) − N +2) , while the blow-up formula gives this coefficient as ( − N − · a ν + α,i N . Equating the two andusing N = α · α and ν · α ≡ (cid:3) The last step is proved exactly as in [Mu˜n00]; we include the proof anyway for complete-ness.
Lemma 5.10 (Step 5) . Let ( X, ν ) : ( Y , λ ) → ( Y , λ ) be a cobordism with b ( X ) = 0 , andlet Σ ⊂ X be a smoothly embedded surface of genus g ≥ and positive self-intersection.Then | K (Σ) | + Σ · Σ ≤ g (Σ) − for every basic class K of X .Proof. Let ˜ X be the N -fold blow-up of X with exceptional spheres e , . . . , e N , where N =Σ · Σ >
0; and let ˜Σ ⊂ ˜ X denote the proper transform Σ − P Nj =1 e j . We let ˜ ν be whicheverof ν and ν + e satisfies ˜ ν · ˜Σ ≡ · ˜Σ = 0, so we can apply Lemma 5.5 to D ˜ ν ˜ X and ˜Σ to see that | ˜ K ( ˜Σ) | ≤ g − K of ˜ X . These basic classes include K ± P Nj =1 E j , and since K ± N X j =1 E j ( ˜Σ) = K (Σ) ± N = K (Σ) ± Σ · Σwe must have | K (Σ) | + Σ · Σ ≤ g − (cid:3) Lemmas 5.7, 5.8, 5.9, and 5.10 collectively prove Proposition 5.4. (cid:3)
Injective maps induced by trace cobordisms.
Recall that the trace of n -surgeryon a framed knot K ⊂ S is the cobordism X n ( K ) : S → S n ( K )obtained from S × [0 ,
1] by attaching an n -framed 2-handle along K × { } . In this section,we prove that the map on framed instanton homology induced by the trace of 1-surgery onthe torus knot T , is injective; see Proposition 5.13. We will use this in the next section toestablish the general case of Theorem 5.1, for cobordisms with b +2 ( X ) = 0. We start with some lemmas about the dimension of I for surgeries on T , . Lemma 5.11. dim I ( S k ( T , )) = k for all k ≥ .Proof. Observe that 9-surgery on T , is a lens space of order 9 [Mos71], and hencedim I ( S ( T , )) = 9 . Using (2.6), we have an exact triangle of the form · · · → I ( S ) I ( X k ( T , ) ,ν k ) −−−−−−−−−−→ I ( S k ( T , )) → I ( S k +1 ( T , )) → . . . for all k ≥
1. Each X k ( T , ) contains a surface Σ k of genus g ( T , ) = 2 and self-intersection k , built by gluing a Seifert surface for T , to a core of the k -framed 2-handle, and when k > I ( X k ( T , ) , ν k ) to vanish by the adjunction inequality of Lemma 5.10.Thus dim I ( S k +1 ( T , )) = dim I ( S k ( T , )) + 1for all k ≥
3, and the lemma follows by induction. (cid:3)
Lemma 5.12. dim I ( S ( T , )) = 5 .Proof. We first note that S ( T , ) ∼ = − Σ(2 , , . Since the dimension of I does not change upon reversing orientation, it suffices to provethat I (Σ(2 , , §
6] that, with coefficientsin Z , I ∗ (Σ(2 , , ∼ = Z ⊕ Z ⊕ Z ⊕ Z , where the subscripts denote the (absolute, in this case) Z / Z grading. Frøyshov computedin [Frø04, Proposition 1] that h (Σ(2 , , , where h is his invariant from [Frø02]. The h invariant is defined by2 h ( Y ) = χ ( I ∗ ( Y )) − χ ( ˆ I ∗ ( Y ))for any homology 3-sphere Y , where ˆ I ∗ is reduced instanton homology [Frø02], and so χ ( ˆ I ∗ (Σ(2 , , . Reduced instanton homology satisfiesrank ˆ I q ( Y ) ≤ rank I q ( Y )for all q , by definition, with equality if q ≡ , q I q ( Y ) is asubgroup of I q ( Y ) (see [Frø02, § I ∗ (Σ(2 , , ∼ = V ⊕ Z ⊕ V ⊕ Z for some V , V ⊂ Z . Moreover, there is a degree-4 endomorphism u on ˆ I ∗ ( Y ) such that u −
64 is nilpotent [Frø02, Theorem 10], which implies that u : V → V is an isomorphism.From χ ( ˆ I ∗ (Σ(2 , , I ∗ (Σ(2 , , ∼ = Z ⊕ Z ⊕ Z ⊕ Z is free of rank one in each even grading. NSTANTONS AND L-SPACE SURGERIES 33
We now apply a corollary by Scaduto [Sca15, Corollary 1.5] of Fukaya’s connected sumtheorem, which says that since Y = Σ(2 , ,
9) is ± C , I ( Y ) ∼ = H ∗ (pt; C ) ⊕ H ∗ ( S ; C ) ⊗ M j =0 ˆ I j ( Y )as Z / Z -graded C -modules. Thus, with C -coefficients, we have I (Σ(2 , , ∼ = C ⊕ C ⊕ C ⊕ C , which completes the proof. (cid:3) Finally, from (2.6) we have an exact triangle · · · → I ( S ) → I ( S ( T , )) → I ( S ( T , )) → . . . , which together with Lemma 5.11 implies thatdim I ( S ( T , )) ≤ . Similarly, from Lemma 5.12 and the exact triangle(5.2) · · · → I ( S ) I ( X ( T , ) ,ν ) −−−−−−−−−−→ I ( S ( T , )) → I ( S ( T , )) → . . . , we deduce that dim I ( S ( T , )) ≥ , hence the dimension must be equal to 4. Since equality holds iff the map I ( X ( T , ) , ν )is injective, we have proved the following. Proposition 5.13.
The map I ( X ( T , ) , ν ) : I ( S ) → I ( S ( T , )) in (5.2) is injective,where X ( T , ) is the trace of 1-surgery on T , . (cid:3) The structure theorem in general.
We now deduce the general case of Theorem 5.1from the case where b +2 ( X ) is positive. In this subsection we take( X, ν ) : ( Y , λ ) → ( Y , λ )to be a cobordism with b ( X ) = 0 but with no restrictions on b +2 ( X ).We begin by taking a 3-ball in Y which avoids λ , identifying a T , knot inside this ball,and letting Y ′ = Y S ( T , )denote the result of 1-surgery along this T , . The trace of this surgery is a cobordism( Z, λ × [0 , ∪ ν ) : ( Y , λ ) → ( Y ′ , λ ) . built by attaching a 2-handle to Y × [0 ,
1] along T , × { } , where ν is the cobordism ofProposition 5.13 suitably interpreted. Lemma 5.14.
The induced map I ( Z, λ × [0 , ∪ ν ) : I ( Y , λ ) → I ( Y ′ , λ ) is injective. Proof.
Letting X n ( K ) : S → S n ( K ) be the trace of n -surgery on K , as in § Z ∼ = ( Y × [0 , ⊲⊳ X ( T , ) . The K¨unneth isomorphism I ( Y Y ′ , λ + λ ′ ) ∼ = −→ I ( Y, λ ) ⊗ I ( Y ′ , λ ′ )is natural with respect to split cobordisms [Sca15, § I ( Y S ) I (( Y × [0 , ⊲⊳X ( T , )) / / ∼ = (cid:15) (cid:15) I ( Y S ( T , )) ∼ = (cid:15) (cid:15) I ( Y ) ⊗ I ( S ) Id ⊗ I ( X ( T , )) / / I ( Y ) ⊗ I ( S ( T , ))commutes. (We omit the various λ , λ × [0 , ν from the diagram for readability.)The map I ( X ( T , ) , ν ) is injective by Proposition 5.13, so it follows that the top arrow I ( Z, λ × [0 , ∪ ν ) is injective as well. (cid:3) We now modify the cobordism X by attaching Z to get( X ′ , ν ′ ) = ( X, ν ) ∪ ( Y ,λ ) ( Z, λ × [0 , ∪ ν ) : ( Y , λ ) → ( Y ′ , λ ) . Then b ( X ′ ) = b ( X ) = 0, and H ( X ′ ) ∼ = H ( X ) ⊕ H ( Z ), where H ( Z ) ∼ = Z is generatedby a surface F built by gluing a Seifert surface for T , to the core of the 2-handle. LetΣ , . . . , Σ k be an integral basis of H ( X ; Z ) / torsion. Since F · F = 1, we have b +2 ( X ′ ) > K ′ , . . . , K ′ r : H ( X ′ ) → Z and homomorphisms a ′ , . . . , a ′ r : I ( Y , λ ) → I ( Y ′ , λ )such that D ν ′ X ′ ( sF + t Σ + · · · + t k Σ k ) = e Q ( sF + P i t i Σ i ) / r X j =1 a ′ j e sK ′ j ( F )+ P i t i K ′ j (Σ i ) . For convenience, we will work with modified Donaldson series as in [FS95], namely K ν ′ X ′ = e − Q X ′ / D ν ′ X ′ and K νX = e − Q X / D νX , so that(5.3) K ν ′ X ′ ( sF + t Σ + · · · + t k Σ k ) = r X j =1 a ′ j e sK ′ j ( F )+ P i t i K ′ j (Σ i ) . Letting S = t Σ + · · · + t k Σ k , we observe that K ν ′ X ′ ( S ) = e − Q X ′ ( S ) / D X ′ ,ν ′ ( − ⊗ (1 + x ) e S )= D Z,λ × [0 , ∪ ν ( − ⊗ ◦ e − Q X ( S ) / D X,ν ( − ⊗ (1 + x ) e S )or equivalently(5.4) K ν ′ X ′ ( t Σ + · · · + t k Σ k ) = I ( Z, λ × [0 , ∪ ν ) ◦ K νX ( t Σ + · · · + t k Σ k )as formal series in the variables t i . NSTANTONS AND L-SPACE SURGERIES 35
Lemma 5.15.
For fixed ( X, ν ) , there are finitely many basic classes K ν,i : H ( X ; Z ) → Z and nonzero maps a ν,i ∈ Hom( I ( Y , λ ) , I ( Y , λ )) , each with rational coefficients, such that D νX ( h ) = e Q ( h ) / X i a ν,i e K ν,i ( h ) . Proof.
For each i = 1 , . . . , k , we define a finite set A i = { K ′ j (Σ i ) | ≤ j ≤ r } ⊂ Z and an operator δ i = Y c ∈ A i (cid:18) ∂∂t i − c (cid:19) , and it follows immediately from equation (5.3) that δ i K ν ′ X ′ ( sF + t Σ + · · · + t k Σ k ) = 0 . Setting s = 0, we deduce from equation (5.4) that I ( Z ; λ × [0 , ∪ ν ) ◦ δ i K νX ( t Σ + · · · + t k Σ k ) = 0 , and Lemma 5.14 says that I ( Z ; λ × [0 , ∪ ν ) is injective, so in fact δ i K νX ( t Σ + · · · + t k Σ k ) = 0 , i = 1 , . . . , k. This means that K νX = e − Q X / D νK has the form K νX k X i =1 t i Σ i ! = X c =( c ,...,c k ) ∈ A ×···× A k a c e c t + ··· + c k t k for some homomorphisms a c , which have rational coefficients just as in the case b +2 ( X ) > (cid:3) For the following lemma, given a homomorphism K ′ : H ( X ′ ; Z ) → Z , we will write K ′ | X to denote the composition H ( X ; Z ) i ∗ −→ H ( X ′ ; Z ) K ′ −−→ Z , in which i : X ֒ → X ′ is the obvious inclusion. Lemma 5.16. If K is a basic class for ( X, ν ) , then there is a basic class K ′ for ( X ′ , ν ′ ) such that K = K ′ | X .Proof. We take A i = { K ′ j (Σ i ) } for i = 1 , . . . , k as before, and we recall from the proof ofLemma 5.15 that every basic class of ( X, ν ) has the form K ν,j ( t Σ + · · · + t k Σ k ) = c t + · · · + c k t k for some integers c i ∈ A i . We thus define an operator(5.5) δ K = k Y i =1 Y c ∈ A i c = K (Σ i ) ∂∂t i − cK (Σ i ) − c ! , and it follows that δ K e K ν,j ( t Σ + ··· + t k Σ k ) = ( e K ( t Σ + ··· + t k Σ k ) K ν,j = K K ν,j = K. In particular, if K is a basic class of ( X, ν ) with coefficient a = 0 in K νX , then δ K K νX ( t Σ + · · · + t k Σ k ) = ae K ( t Σ + ··· + t k Σ k ) = 0 , and so by Lemma 5.14 and equation (5.4) we see that δ K K ν ′ X ′ ( t Σ + · · · + t k Σ k ) = r X j =1 a ′ j · δ K e K ′ j ( t Σ + ··· + t k Σ k ) is nonzero as well, where the equality follows from (5.3). Each K ′ j (Σ i ) belongs to the set A i , so if the j th term is nonzero for some fixed j then again we must have K ′ j (Σ i ) = K (Σ i )for all i , and so K ′ j | X = K . (cid:3) Lemma 5.17.
The basic classes K for ( X, ν ) satisfy K ( h ) + h · h ≡ for all h ∈ H ( X ; Z ) .Proof. Write K = K ′ | X for some basic class K ′ on X ′ , which we can do by Lemma 5.16.Then since b + ( X ′ ) >
0, we may apply Proposition 5.4 to X ′ to conclude that K ( h ) + h · h = K ′ ( h ) + h · h ≡ . (cid:3) Lemma 5.18.
Let K ν,i be the basic classes for ( X, ν ) , with coefficients a ν,i , so that D νX ( h ) = e Q ( h ) / X i a ν,i e K ν,i ( h ) . Then for all α ∈ H ( X ; Z ) , we have D ν + αX ( h ) = e Q ( h ) / X i ( − ( K ν,i ( α )+ α · α )+ ν · α · a ν,i e K ν,i ( h ) . In particular, the set of basic classes does not depend on ν .Proof. Fix α ∈ H ( X ; Z ) and let K be a basic class for ( X, ν ), with coefficient a ν = 0 in D νX and coefficient a ν + α (possibly zero) in D ν + αX . We write ǫ α = ( − ( K ( α )+ α · α )+ ν · α , and set S = t Σ + · · · + t k Σ k for readability.Recalling that K νX = e − Q X / D νX and likewise for K ν ′ X ′ , we now observe by (5.4) that I ( Z, λ × [0 , ∪ ν ) ◦ δ K ( K ν + αX ( S ) − ǫ α K νX ( S )) = δ K ( K ν ′ + αX ′ ( S ) − ǫ α K ν ′ X ′ ( S )) , where δ K is the operator defined in (5.5). On the right side, we have δ K K ν ′ + αX ′ ( S ) − ǫ α δ K K ν ′ X ′ ( S ) = X K ′ j | X = K ( a ′ ν ′ + α,j − ǫ α a ′ ν ′ ,j ) e K ′ j ( S ) , NSTANTONS AND L-SPACE SURGERIES 37 and each of the coefficients a ′ ν ′ + α,j − ǫa ′ ν ′ ,j is zero by Lemma 5.9 and the fact that K ′ j ( α ) = K ( α ), so this vanishes. On the left side, Lemma 5.14 says that I ( Z, λ × [0 , ∪ ν ) isinjective, so we deduce that δ K ( K ν + αX ( S ) − ǫ α K νX ( S )) = a ν + α e K ( S ) − ǫ α a ν e K ( S ) is zero as well. Thus a ν + α = ǫ α a ν and the lemma follows. (cid:3) Proof of Theorem 5.1.
The case where b +2 ( X ) > b +2 ( X ) = 0then the adjunction inequality is vacuously true, while the rest of the theorem is a combi-nation of Lemmas 5.15, 5.17, and 5.18. (cid:3) A decomposition of cobordism maps
The goal of this section is to use the structure theorem of the previous section (Theorem5.1) to prove Theorem 1.16, which extends the eigenspace decomposition of framed instantonhomology in Theorem 2.25 to a similar decomposition for cobordism maps, akin to the Spin c decompositions of cobordism maps in Heegaard and monopole Floer homology.We will assume throughout this section that( X, ν ) : ( Y , λ ) → ( Y , λ )is a cobordism with b ( X ) = 0. Recall that Theorem 1.16 says that there is a decompositionof the induced cobordism map as a sum I ( X, ν ) = X s : H ( X ; Z ) → Z I ( X, ν ; s )of maps I ( X, ν ; s ) : I ( Y , λ ; s | Y ) → I ( Y , λ ; s | Y )which satisfy five properties. The proof of this theorem below will make reference to theseproperties as they are numbered in the theorem statement. Proof of Theorem 1.16.
By Theorem 5.1, we can write(6.1) D νX ( h ) = e Q ( h ) / r X j =1 a ν,j e K j ( h ) , where the a j are nonzero homomorphisms I ( Y , λ ) → I ( Y , λ ) and the basic classes K j are elements of Hom( H ( X ; Z ) , Z ). Then for any homomorphism s : H ( X ; Z ) → Z , wedefine I ( X, ν ; s ) : I ( Y , λ ) → I ( Y , λ )to be a ν,j if K j = s for some j , and I ( X, ν ; s ) = 0 otherwise.Properties (1), (2), and (5) are immediate from the definition of I ( X, ν ; s ) and Theo-rem 5.1, since I ( X, ν ; s ) is nonzero precisely when s is a basic class. Property (4) followsfrom the blow-up formula for D νX , Theorem 5.3. It remains to verify the following: • The identity I ( X, ν ) = P s I ( X, ν ; s ); • The fact that each I ( X, ν ; s ) is zero on all I ( Y , λ ; s ) except for s = s | Y , andthat its image lies in I ( Y , λ ; s | Y ); • Property (3), the composition law.
For the first of these, we set h = 0 in (6.1) to get(6.2) D X,ν (cid:0) − ⊗ (cid:0) x (cid:1)(cid:1) = r X j =1 a ν,j = 2 X s : H ( X ; Z ) → Z I ( X, ν ; s ) . But x = µ (pt) acts on I ( Y , λ ) as multiplication by 2, as in Remark 2.17, so for all a ∈ I ( Y , λ ) we have D X,ν (cid:0) a ⊗ (cid:0) x (cid:1)(cid:1) = D X,ν (cid:16)(cid:16) µ (pt)2 (cid:17) a ⊗ (cid:17) = I ( X, ν )(2 a ) , and hence the left side of (6.2) is 2 I ( X, ν ), establishing the claim. The second and thirditems above are proved below as Proposition 6.3 and Proposition 6.5, respectively. (cid:3)
Remark 6.1. If I ( X, ν ; s ) is nonzero, then s | Y must take values in 2 Z , since for any class h ∈ H ( Y ; Z ) we have s | Y ( h ) = s (( i ) ∗ h ) ≡ ( i ) ∗ h · ( i ) ∗ h = h · h = 0 (mod 2)by property (2) of Theorem 1.16, which we have already proved. Remark 6.2.
Properties (2) and (5) of Theorem 1.16 imply that I ( X, ν + 2 α ; s ) = ( − α · α I ( X, ν ; s )for all s . Indeed, both sides are zero unless s ( α ) ≡ α · α (mod 2), and if they are nonzerothen by (5) the exponent on the right should be ( s (2 α ) + (2 α ) · (2 α )) + ν · (2 α ) ≡ s ( α ) ≡ α · α (mod 2) . Proposition 6.3.
We have the following: • For any s : H ( Y ; Z ) → Z , the map I ( X, ν ; s ) is zero on I ( Y , λ ; s ) unless s = s | Y . • The image of I ( X, ν ; s ) lies in I ( Y , λ ; s | Y ) .In other words, for each s we can interpret I ( X, ν ; s ) as a map I ( X, ν ; s ) : I ( Y , λ ; s | Y ) → I ( Y , λ ; s | Y ) . Proof.
We fix a non-torsion class Σ ∈ H ( Y ; Z ) and extend it to a rational basis Σ =Σ , Σ , . . . , Σ k of H ( X ; Z ). (The inclusion Y = ∂X ֒ → X gives an injection H ( Y ; Q ) → H ( X ; Q ), by the long exact sequence of the pair ( X, Y ) and the fact that H ( X, Y ; Q ) ∼ = H ( X ; Q ) = 0.) We then write D νX ( t Σ + · · · + t k Σ k ) = e Q ( P t i Σ i ) / r X j =1 a j e K j ( t Σ + ··· + t k Σ k ) by Theorem 5.1, and observe that Q k X i =1 t i Σ i ! = Q k X i =2 t i Σ i ! does not depend on t , since H ( Y ) is in the kernel of the intersection pairing on X . NSTANTONS AND L-SPACE SURGERIES 39
We now take an arbitrary element z ∈ I ( Y , λ ; s ) and an integer n ≥
1, and we applythe operator (cid:16) ∂∂t − s (Σ ) (cid:17) n to both sides of D νX ( t Σ + · · · + t k Σ k )( z ) = D X,ν (cid:0) z ⊗ (cid:0) x (cid:1) e t Σ + ··· + t k Σ k (cid:1) . On the left side, we have(6.3) (cid:18) ∂∂t − s (Σ ) (cid:19) n D νX ( t Σ + · · · + t k Σ k )( z )= e Q ( P t i Σ i ) / r X j =1 a j ( z ) ( K j (Σ ) − s (Σ )) n e K j ( t Σ + ··· + t k Σ k ) . On the right side, we have D X,ν (cid:0) z ⊗ (cid:0) x (cid:1) (Σ − s (Σ )) n e t Σ + ··· + t k Σ k (cid:1) = D X,ν (cid:0) ( µ (Σ ) − s (Σ )) n z ⊗ (cid:0) x (cid:1) e t Σ + ··· + t k Σ k (cid:1) , which is equivalently(6.4) (cid:0) D νX ( t Σ + · · · + t k Σ k ) (cid:1)(cid:0) ( µ (Σ ) − s (Σ )) n z (cid:1) = e Q ( P t i Σ i ) / r X j =1 a j (cid:0) ( µ (Σ ) − s (Σ )) n z (cid:1) e K j ( t Σ + ··· + t k Σ k ) . The various functions e K j ( t Σ + ··· + t k Σ k ) for 1 ≤ j ≤ r are linearly independent as powerseries in t , . . . , t k , so they must have the same coefficients in (6.3) and (6.4), i.e., a j ( z ) · ( K j (Σ ) − s (Σ )) n = a j (cid:0) ( µ (Σ ) − s (Σ )) n z (cid:1) . The right side is identically zero for n large, since z by definition belongs to the (generalized) s (Σ )-eigenspace of µ (Σ ). Thus the left side is zero for n large as well, which implies thatit is also zero when n = 1, so in fact we have(6.5) a j ( z ) · ( K j (Σ ) − s (Σ )) = a j (cid:0) ( µ (Σ ) − s (Σ )) z (cid:1) = 0for all j . We conclude that if K j (Σ ) = s (Σ ) then a j ( z ) = 0, and hence I ( X, ν ; s )( z ) = 0for all s which do not satisfy s (Σ ) = s (Σ ). But Σ = Σ was an arbitrary non-torsionclass in H ( Y ; Z ), so if I ( X, ν ; s )( z ) = 0 then s | Y = s on all of H ( Y ; Z ).The proof that I ( X, ν ; s ) sends I ( Y , λ ) into I ( Y , λ ; s | Y ) is similar. We chooseour basis { Σ k } so that Σ is a given non-torsion element of H ( Y ; Z ), and then apply ∂∂t to both sides of e Q ( P t i Σ i ) / r X j =1 a j ( z ) e K j ( t Σ + ··· + t k Σ k ) = D X,ν (cid:0) z ⊗ (cid:0) x (cid:1) e t Σ + ··· + t k Σ k (cid:1) to get e Q ( P t i Σ i ) / r X j =1 K j (Σ ) · a j ( z ) e K j ( t Σ + ··· + t k Σ k )0 JOHN A. BALDWIN AND STEVEN SIVEK on the left side, and D X,ν (cid:0) z ⊗ (cid:0) x (cid:1) Σ e t Σ + ··· + t k Σ k (cid:1) = µ (Σ ) D X,ν (cid:0) z ⊗ (cid:0) x (cid:1) e t Σ + ··· + t k Σ k (cid:1) = µ (Σ ) · ( D νX ( t Σ + · · · + t k Σ k ))( z )= e Q ( P t i Σ i ) / r X j =1 µ (Σ ) a j ( z ) e K j ( t Σ + ··· + t k Σ k ) on the right. Again, the linear independence of the e K j ( P t i Σ i ) tells us that(6.6) µ (Σ ) a j ( z ) = K j (Σ ) a j ( z )for all j , so if I ( X, ν ; s )( z ) is nonzero, then s = K j for some j and we have I ( X, ν ; s )( z ) = 12 a j ( z ) ∈ ker( µ (Σ ) − s (Σ )) . Since Σ was an arbitrary non-torsion element of H ( Y ; Z ), we conclude that the image of I ( X, ν ; s ) lies in \ h ∈ H ( Y ; Z ) ker( µ ( h ) − s ( h )) ⊂ I ( Y , λ ; s | Y ) . (cid:3) Remark 6.4.
We can deduce from the proof of Proposition 6.3, and in particular equa-tion (6.5), that for all z ∈ I ( Y , λ ; s ) and all Σ ∈ H ( Y ; Z ) we have( µ (Σ) − s (Σ)) z ∈ ker I ( X, ν ; s )for all s . Similarly, equation (6.6) shows that the image of a map I ( X, ν ; s ) consists ofactual simultaneous eigenvectors, as opposed to generalized ones. Proposition 6.5.
Let ( X, ν ) denote the composite cobordism ( Y , λ ) ( X ,ν ) −−−−−−→ ( Y , λ ) ( X ,ν ) −−−−−−→ ( Y , λ ) , where b ( X ) = b ( X ) = b ( X ) = 0 . Then for all homomorphisms s : H ( X ; Z ) → Z and s : H ( X ; Z ) → Z , we have I ( X , ν ; s ) ◦ I ( X , ν ; s ) = X s : H ( X ; Z ) → Z s | X = s , s | X = s I ( X, ν ; s ) . Proof.
Let Σ , . . . , Σ k and Σ ′ , . . . , Σ ′ k ′ be integral bases of H ( X ; Z ) and H ( X ; Z ), re-spectively, modulo torsion, and write D ν X = e Q/ r X j =1 a j e K j , D ν X = e Q/ r ′ X j =1 a ′ j e K ′ j . We define some finite sets of integers by A i = { K j (Σ i ) | ≤ j ≤ r } , A ′ i = { K ′ j (Σ ′ i ) | ≤ j ≤ r ′ } and let d i = s (Σ i ) and d ′ i = s (Σ ′ i ) for all i ; here i ranges from 1 to k for A i and d i , andfrom 1 to k ′ for A ′ i and d ′ i . Then we define some differential operators by δ s = k Y i =1 Y c ∈ A i r { s (Σ i ) } ∂∂t i − cd i − c ! , δ s = k ′ Y i =1 Y c ∈ A ′ i r { s (Σ ′ i ) } ∂∂t ′ i − cd ′ i − c ! NSTANTONS AND L-SPACE SURGERIES 41 so that if we write K = e − Q/ D for each cobordism then δ s K ν X ( t Σ + · · · + t k Σ k ) = ( a j e K j ( t Σ + ··· + t k Σ k ) s = K j s
6∈ { K , . . . , K r } . Upon setting ( t , . . . , t k ) = (0 , . . . , δ s K ν X ( t Σ + · · · + t k Σ k ) (cid:12)(cid:12)(cid:12) t = ··· = t k =0 = 2 I ( X , ν ; s ) , and the same argument gives δ s K ν X ( t ′ Σ ′ + · · · + t ′ k ′ Σ ′ k ′ ) (cid:12)(cid:12)(cid:12) t ′ = ··· = t ′ k ′ =0 = 2 I ( X , ν ; s ) . Letting S = t Σ + · · · + t k Σ k and S ′ = t ′ Σ ′ + · · · + t ′ k ′ Σ ′ k ′ for convenience, we nowcompute D ν X ( S ′ ) ◦ D ν X ( S ) = D X ,ν (cid:0) − ⊗ (cid:0) x (cid:1) e S (cid:1) ◦ D X ,ν (cid:16) − ⊗ (cid:0) x (cid:1) e S ′ (cid:17) = D X,ν (cid:16) − ⊗ (cid:0) x (cid:1) e S + S ′ (cid:17) = 2 D νX ( S + S ′ ) , since the extra factor of 1 + x in the second line acts on I ( Y , λ ) as multiplication by 2.We have S · S ′ = 0 and hence Q X ( S + S ′ ) = Q X ( S ) + Q X ( S ′ ), so dividing both sides by4 exp( Q X ( S + S ′ ) /
2) gives K ν X ( S ′ ) ◦ K ν X ( S ) = K νX ( S + S ′ ) . Now applying δ s ◦ δ s and then setting all t i and t ′ i to zero turns the left side into I ( X , ν ; s ) ◦ I ( X , ν ; s ) . On the right side, the operator δ s ◦ δ s fixes all terms in the series K νX ( S + S ′ ) of theform a · e K ( S + S ′ ) , where the basic class K satisfies K (Σ i ) = s (Σ i ) , ≤ i ≤ k and K (Σ ′ i ) = s (Σ ′ i ) , ≤ i ≤ k ′ , or equivalently K | X = s and K | X = s ; and it replaces all other terms with 0. Settingthe t i and t ′ i to zero then gives the sum of I ( X, ν ; s ) over all s such that s | X = s and s | X = s , as desired. (cid:3) Weaker versions of Proposition 6.5 still hold in the case where b ( X ) = 0 but one of b ( X ) and b ( X ) is positive, even though we have not proved a structure theorem forthe corresponding cobordism map on I . We include one such statement for completeness. Proposition 6.6.
Let ( X, ν ) denote the composite cobordism ( Y , λ ) ( X ,ν ) −−−−−−→ ( Y , λ ) ( X ,ν ) −−−−−−→ ( Y , λ ) , where b ( X ) = b ( X ) = 0 and X is a rational homology cobordism. Then I ( X, ν ; s ) = I ( X , ν ) ◦ I ( X , ν ; s | X ) for all homomorphisms s : H ( X ; Z ) → Z . Proof.
We take a rational basis Σ , . . . , Σ k of H ( X ; Z ), which by hypothesis is also arational basis of H ( X ; Z ). Then, following the proof of Proposition 6.5, we can write D νX ( t Σ + · · · + t k Σ k ) = D X ,ν ( − ⊗ ◦ D ν X ( t Σ + · · · + t k Σ k )and divide both sides by e Q ( P i t i Σ i ) / to get K νX ( t Σ + · · · + t k Σ k ) = I ( X , ν ) ◦ K ν X ( t Σ + · · · + t k Σ k ) . We apply the differential operator k Y i =1 Y c ∈ A i c = s (Σ i ) ∂∂t i − cs (Σ i ) − c ! to both sides, where A i is the set of values of K (Σ i ) as K ranges over all basic classes of X , and then set t = · · · = t k = 0 and divide by 2 to get the desired relation. (cid:3) Finally, the adjunction inequality in Theorem 1.16 gives us the following general result.
Proposition 6.7.
Let ( X, ν ) : ( Y , λ ) → ( Y , λ ) be a cobordism with b ( X ) = 0 . Supposethat X contains a smoothly embedded, closed surface S satisfying one of the following: • [ S ] · [ S ] ≥ max(2 g ( S ) − , ; or • S is a sphere with [ S ] · [ S ] = 0 and [ S ] · [ F ] = 0 for some smooth, closed surface F ⊂ X .Then the cobordism map I ( X, ν ) is identically zero.Proof. In the first case, if g ( S ) ≥ I ( X, ν ) is a sum of variousmaps I ( X, ν ; s ), all of which are zero by part (2) of that theorem. If instead S is a spherethen we attach a handle to get a torus T of positive self-intersection and apply the sameargument to T .In the second case, we follow [KM95b, § F with − F if needed, we arrangethat n = [ S ] · [ F ] is positive and that S ∩ F consists of n points. We form another surface S ′ by taking the union of F with d > S and smoothing out the dn points of intersection. The result has genus g ( S ′ ) = 1 − χ ( S ′ )2 = 1 − (2 − g ( F )) + 2 d − dn g ( F ) + d ( n − S ′ ] · [ S ′ ] = ([ F ] + d [ S ]) · ([ F ] + d [ S ]) = [ F ] · [ F ] + 2 dn. For d sufficiently large, we have [ S ] · [ S ] ≥
1, and also[ S ′ ] · [ S ′ ] − (2 g ( S ′ ) −
1) = 2 d + ([ F ] · [ F ] − (2 g ( F ) − > , so we can apply the first case to S ′ to conclude that I ( X, ν ) = 0. (cid:3)
NSTANTONS AND L-SPACE SURGERIES 43 Instanton L-space knots are fibered
Our primary goal in this section is to prove that instanton L-space knots are fibered, withSeifert genus equal to smooth slice genus (Theorem 7.8). The equality g = g s of Seifert andsmooth slice genus will also follow from our result, proved in the next section, that instantonL-space knots are strongly quasipositive. However, our proof of strong quasipositivity usesthe Giroux correspondence, whereas our proof that g = g s here does not. We will alsoprove in this section some results related to the bound on L-space slopes in Theorem 1.15,which are enough to conclude, without using the Giroux correspondence, that the framedinstanton homology of surgeries detects the trefoils among nontrivial knots. We will provethe general 2 g − K ⊂ S is a nontrivial knot of genus g = g ( K ) >
0, with minimal genus Seifert surface Σ. For each integer k ≥
0, consider the 2-handlecobordisms X k : S → S k ( K ) W k +1 : S k ( K ) → S k +1 ( K ) , where X k is the trace of k -surgery on K , and W k +1 is the trace of − K in S k ( K ). A handleslide reveals that(7.1) X k ∪ S k ( K ) W k +1 ∼ = X k +1 CP . By Theorem 2.27 and (2.6), the maps induced by X k and W k +1 fit into exact triangles(7.2) · · · → I ( S ) I ( X ,ν ) −−−−−−→ I ( S ( K ) , µ ) I ( W ,ω ) −−−−−−−→ I ( S ( K )) → . . . and(7.3) · · · → I ( S ) I ( X k ,ν k ) −−−−−−→ I ( S k ( K )) I ( W k +1 ,ω k +1 ) −−−−−−−−−−→ I ( S k +1 ( K )) → . . . for k ≥
1, for some ν k and ω k +1 . Observe that the maps induced by X k and W k +1 shift themod 2 grading by 1 and 0, respectively, for all k ≥
0, by Proposition 2.21.Let us denote by Σ k ⊂ X k the surface of genus g and self-intersection k obtained by gluing a core of the 2-handle usedto form X k to the minimal genus Seifert surface Σ for K . The construction in [Sca15, § ν k ] · [Σ k ] ≡ ( k = 0 k k ≥ . A homomorphism s : H ( X k ; Z ) → Z is determined by its evaluation on [Σ k ], and Theorem 1.16 says that the map I ( X k , ν k ; s )is nonzero only if s ([Σ k ]) ≡ Σ k · Σ k = k (mod 2) . With this as motivation, we denote by t k,i : H ( X k ; Z ) → Z the unique homomorphism sending [Σ k ] to 2 i − k . The adjunction inequality of Theorem 1.16implies that for k ≥ I ( X k , ν k ; t k,i ) : I ( S ) → I ( S k ( K ))is nontrivial only if(7.5) | t k,i ([Σ k ]) | + k ≤ g − ⇐⇒ − g + k ≤ i ≤ g − . For each integer k ≥ i , let us define(7.6) z k,i := I ( X k , ν k ; t k,i )( ) , where is a fixed generator of I ( S ) ∼ = C . For k ≥
1, these elements belong to I ( S k ( K ))since the map induced by X k shifts the mod 2 grading by 1, as noted above. Moreover, for k = 0, we have that z ,i ∈ I ( S ( K ) , µ ; s i )where s i : H ( S ( K ); Z ) → Z is the unique homomorphism sending [ ˆΣ] to 2 i , for ˆΣ the capped-off Seifert surface in S ( K )(note that [ ˆΣ] = [Σ ] ∈ H ( X ; Z )). For convenience, we define y i := z ,i ∈ I ( S ( K ) , µ ; s i ) . Note that y i is nonzero only if 1 − g ≤ i ≤ g − , by the adjunction inequality of Theorem 2.25. As outlined in the introduction, our proofthat instanton L-space knots are fibered relies on understanding the kernel of a compositionof the W k maps in terms of these y i (see Lemma 7.5). We prove the technical results wewill need for this in the subsection below.7.1. The W k cobordism maps. Our goal in this section is to understand the images ofthe elements z k,i defined in (7.6) for k ≥ I ( W , ω ) : I ( S ( K ) , µ ) → I ( S ( K )) I ( W k +1 , ω k +1 ) : I ( S k ( K )) → I ( S k +1 ( K )) . Our main result along these lines is the following.
Proposition 7.1.
There are constants ǫ k = ± for all k ≥ , depending only on k , suchthat I ( W , ω )( z ,i ) = ǫ · ( − i ( z ,i + z ,i +1 ) and also I ( W k +1 , ω k +1 )( z k,i ) = ǫ k z k +1 ,i − z k +1 ,i +1 ) for all k ≥ . This proposition will follow from two lemmas below. To begin, recall from (7.1) that X k ∪ S k ( K ) W k +1 ∼ = X k +1 CP . Furthermore, letting E ⊂ CP be the exceptional sphere, this diffeomorphism identifies [Σ k ]on the left with [Σ k +1 ] − [ E ] on the right as elements of H ( X k +1 CP ) ∼ = Z . NSTANTONS AND L-SPACE SURGERIES 45
Lemma 7.2. H ( W k +1 ; Z ) ∼ = Z , and this group is generated by the class of a surface F k +1 such that [ F k +1 ] = [Σ k +1 ] − ( k + 1)[ E ] in H ( X k +1 CP ; Z ) , and hence F k +1 · F k +1 = − k ( k + 1) .Proof. When k = 0, we can see that H ( W ; Z ) ∼ = Z by turning W upside-down so thatit is obtained from the homology sphere − S ( K ) by attaching a 0-framed 2-handle. Theinclusions S ( K ) ֒ → X and S ( K ) ֒ → W induce isomorphisms Z → Z on second homology,so the generators [Σ ] and [ F ] are both the images of a generator of H ( S ( K )), hence theyagree up to sign in H ( X CP ). We can therefore take [ F ] = [Σ ] = [Σ ] − [ E ].Let us assume from here on that k ≥
1. The Mayer–Vietoris sequence for X k ∪ W k +1 ,together with the fact that H ( S k ( K ); Z ) = 0, gives an exact sequence(7.7) 0 → H ( X k ) ⊕ H ( W k +1 ) → H ( X k +1 CP ) → Z /k Z . A generator [ F k +1 ] of H ( W k +1 ) must be orthogonal to [Σ k ] = [Σ k +1 ] − [ E ] in H ( X k +1 CP )since they can be represented by disjoint surfaces, the former in W x +1 and the latter in X k .Thus, [ F k +1 ] must be an integer multiple of [Σ k +1 ] − ( k + 1)[ E ], which has self-intersection − k ( k + 1). Now, the intersection forms on H ( X k ) ⊕ H ( W k +1 ) and H ( X k +1 CP ) haveGram matrices (cid:16) k
00 [ F k +1 ] (cid:17) and (cid:0) k +1 00 − (cid:1) , respectively, with determinants [ F k +1 ] · k and − ( k + 1). The former is a sublattice of thelatter of some index I ≤ k , by (7.7). We therefore have that k ( k + 1) · k ≤ | [ F k +1 ] · k | = | I ( − ( k + 1)) | ≤ k ( k + 1) . Since the left and right sides are equal, we must have [ F k +1 ] = − k ( k + 1) as claimed, andup to reversing orientation it follows that [ F k +1 ] = [Σ k +1 ] − ( k + 1)[ E ]. (cid:3) Noting that W k +1 has even intersection form, let us define s k +1 ,j : H ( W k +1 ; Z ) → Z to be the unique homomorphism sending [ F k +1 ] to 2 j . In order to understand the imagesof the z k,i under the map induced by W k +1 , we first describe the composition(7.8) I ( W k +1 , ω k +1 ; s k +1 ,j ) ◦ I ( X k , ν k ; t k,i )in terms of the maps induced by X k +1 , as per the lemma below. Lemma 7.3.
For k = 0 , there is a constant ǫ = ± such that the composition (7.8) isequal to ǫ · ( − i (cid:16) I ( X , ν ; t ,i ) + I ( X , ν ; t ,i +1 ) (cid:17) when j = i , and it is zero for j = i . For k ≥ , the composition (7.8) is instead equal to ( ǫ k I ( X k +1 , ν k +1 ; t k +1 ,i ) j = i − ǫ k I ( X k +1 , ν k +1 ; t k +1 ,i +1 ) j = i − k, where ǫ k = ± depends only on k , and it is zero otherwise. Proof.
We first aim to understand when the composition (7.8) is nonzero. Scaduto showedin [Sca15, § ν k ∪ ω k +1 ) · E is odd in X k ∪ S k ( K ) W k +1 ∼ = X k +1 CP , so, up to homology, we can write ν k ∪ ω k +1 = ν ′ k +1 ∪ a k E for some properly embedded surface ν ′ k +1 ⊂ X k +1 and some odd a k . We then have that[ ν k ] · [Σ k ] = [ ν k ∪ ω k +1 ] · [Σ k ] = [ ν ′ k +1 ∪ a k E ]([Σ k +1 ] − [ E ]) ≡ [ ν ′ k +1 ] · [Σ k +1 ] + 1 (mod 2) . Combining this with (7.4) gives(7.9) [ ν ′ k +1 ∪ a k E ] ≡ ( [ ν ] + [ E ] + [Σ ] k = 0[ ν k +1 ] + [ E ] k ≥ . Theorem 1.16 says that the composition (7.8) is equal to(7.10) X s I ( X k +1 CP , ν ′ k +1 + a k E ; s ) , where we sum over all s : H ( X k +1 CP ; Z ) → Z with s | X k = t k,i and s | W k +1 = s k +1 ,j , meaning that(7.11) s ([Σ k ]) = 2 i − k and s ([ F k +1 ]) = 2 j. Moreover, properties (2) and (4) of Theorem 1.16 say that each summand is zero unless(7.12) s ([Σ k +1 ]) ≡ k + 1 (mod 2) and s ([ E ]) = ± . The constraints in (7.12) imply that we can write each s which might contribute a nonzerosummand in (7.10) as s = t k +1 ,ℓ + ce, where e is Poincar´e dual to [ E ] and c = ±
1. Using Lemma 7.2, we compute s ([Σ k ]) = s ([Σ k +1 ] − [ E ]) = (2 ℓ − ( k + 1)) + c (7.13) s ([ F k +1 ]) = s ([Σ k +1 ] − ( k + 1)[ E ]) = (2 ℓ − ( k + 1)) + ( k + 1) c. (7.14)By (7.11), this implies that2 j − (2 i − k ) = s ([ F k +1 ]) − s ([Σ k ]) = kc, or, equivalently, that k ( c −
1) = 2( j − i ).Suppose first that k = 0 and c = −
1. Let α k be the class in H ( X k +1 CP ; Z ) satisfying2 α k = [ ν ′ k +1 ∪ a k E ] − ([ ν k +1 ] + [ E ])(such a class exists by (7.9)). Then k = i − j and2 i − k = s ([Σ k ]) = 2( ℓ − − k, where the latter equality is by (7.13), so that ℓ = i + 1. Remark 6.2 and the blow-up formula(property (4) of Theorem 1.16) then say that I ( X k +1 CP , ν ′ k +1 + a k E ; t k +1 ,i +1 − e ) = − ǫ k I ( X k +1 CP , ν k +1 + E ; t k +1 ,i +1 − e )= − ǫ k I ( X k +1 , ν k +1 ; t k +1 ,i +1 ) , NSTANTONS AND L-SPACE SURGERIES 47 where ǫ k = ( − α k · α k +1 . Similarly, if either k = 0 or c = 1 then we must have i = j , andalso 2 i − k = s ([Σ k ]) = (2 ℓ − ( k + 1)) + c = ( ℓ − k c = 12( ℓ − − k ( k, c ) = (0 , − . Thus, either c = 1 and ℓ = i , or ( k, c ) = (0 , −
1) and ℓ = i + 1. In the first case, if k ≥ I ( X k +1 CP , ν ′ k +1 + a k E ; t k +1 ,i + e ) = ǫ k I ( X k +1 , ν k +1 ; t k +1 ,i ) , and if k = 0 then it is(7.15) I ( X CP , ν ′ + a E ; t ,i + e ) . In the second case the relevant map is(7.16) I ( X CP , ν ′ + a E ; t ,i +1 − e ) . This completes the proof of the lemma except for the signs when k = 0. We let α bethe class in H ( X CP ; Z ) satisfying2 α = [ ν ′ + a E ] − ([ ν ] + [ E ] + [Σ ])(which exists by (7.9)), and set ǫ = ( − α · α . Then I ( X CP , ν ′ + a E ; t ,j ± e ) = ǫ I ( X CP , ν + E + Σ ; t ,j ± e )= ∓ ǫ I ( X , ν + Σ ; t ,j )= ∓ ǫ · ( − σ I ( X , ν ; t ,j )where σ = 12 ( t ,j (Σ ) + Σ · Σ ) + ν · Σ = j + ( ν · Σ ) . Since (7.4) says that ν · Σ ≡ I ( X CP , ν ′ + a E ; t ,j ± e ) = ± ǫ · ( − j I ( X , ν ; t ,j )and so we combine this with equations (7.15) and (7.16) to complete the proof. (cid:3) Proof of Proposition 7.1.
Sum (7.8) over all j , applying Lemma 7.3 to determine each com-position, and evaluate the result on the generator ∈ I ( S ). (cid:3) Instanton L-space knots are fibered with g = g s . Our goal in this section is toprove Theorem 7.8, which asserts that instanton L-space knots are fibered with Seifert genusequal to smooth slice genus. The proposition below is the first of two main ingredients inthe proof of this theorem. Adopting the notation from above, recall that we defined y i := z ,i = I ( X , ν ; t ,i )( ) ∈ I ( S ( K ) , µ ; s i ) , and that I ( S ( K ) , µ ; s i ) = 0 for | i | > g − , by the adjunction inequality of Theorem 2.25. Proposition 7.4. If K ⊂ S as above is a nontrivial instanton L-space knot then I ( S ( K ) , µ ; s i ) = C · y i for all i . In particular, dim I ( S ( K ) , µ ; s i ) = 0 or 1 depending on whether y i = 0 or y i = 0 , respectively. To prepare for the proof of Proposition 7.4, we let ( V k , ¯ ω k ) denote the composition( V k , ¯ ω k ) = ( W k , ω k ) ◦ . . . ( W , ω ) := ( S ( K ) , µ ) → ( S k ( K ) , , which induces the map I ( V k , ¯ ω k ) = I ( W k , ω k ) ◦ · · · ◦ I ( W , ω ) : I ( S ( K ) , µ ) → I ( S k ( K )) , and define for each integer i the element c k,i := I ( V k , ¯ ω k )( y i ) ∈ I ( S k ( K )) . Repeated application of Proposition 7.1 then tells us that2 k c k,i = k X j =0 d j,i z k,i + j , where the coefficient d j,i is a sum of (cid:0) kj (cid:1) signs. In particular, we have(7.17) 2 k c k,i = ± z k,i + k X j =1 d j,i z k,i + j , and since this system of equations has an invertible triangular matrix of coefficients, we seethat each z k,i is a linear combination of the various c k,j . Lemma 7.5.
For all integers n ≥ , the kernel of the map I ( V n , ¯ ω n ) = I ( W n , ω n ) ◦ · · · ◦ I ( W , ω ) : I ( S ( K ) , µ ) → I ( S n ( K )) lies in the span of the elements y − g , . . . , y g − . It is equal to this span if n ≥ g − .Proof. The case n = 1 follows from the exact triangle (7.2), sinceker I ( V , ¯ ω ) = ker I ( W , ω ) = Im I ( X , ν )is spanned by I ( X , ν )( ) = P i y i .Suppose now that n ≥
1. Suppose the kernel of I ( V n , ¯ ω n ) lies in the span of the elements y − g , . . . , y g − . We will deduce that the same is true of the kernel of I ( V n +1 , ¯ ω n +1 ), whichwill then prove the first part of the lemma for all n , by induction.From the exact triangle (7.3), the kernel of I ( W n +1 , ω n +1 ) is generated by I ( X n , ν n )( ) = g − X i =1 − g + n z n,i NSTANTONS AND L-SPACE SURGERIES 49 (here, the range of indices in the sum comes from (7.5)); since each z n,i is a linear combi-nation of the c n,j , as argued above, we may write this element as I ( X n , ν n )( ) = X i z n,i = X j a n,j c n,j = I ( V n , ¯ ω n ) X j a n,j y j for some coefficients a n,j . In particular, any element x of the kernel of I ( V n +1 , ¯ ω n +1 ) = I ( W n +1 , ω n +1 ) ◦ I ( V n , ¯ ω n )satisfies I ( V n , ¯ ω n )( x ) = c · I ( V n , ¯ ω n ) (cid:16)P j a n,j y j (cid:17) for some constant c , and then x − c X j a n,j y j ∈ ker I ( V n , ¯ ω n ) . It follows by induction that x is a linear combination of y − g , . . . , y g − , as desired.Now suppose n ≥ g −
1. It remains to show that I ( V n , ¯ ω n )( y i ) = 0 for all i in this case.We can write I ( V n , ¯ ω n )( y i ) = c n,i as a linear combination of the elements z n,j = I ( X n , ν n ; t n,j )( ) , as before. The adjunction inequality says that the map I ( X n , ν n ; t n,j ) is zero for n ≥ g − z n,j = 0 for all j and hence c n,i = 0 for all i as well. (cid:3) Proof of Proposition 7.4.
Suppose for some i that there is an element x ∈ I ( S ( K ) , µ ; s i )which is not a multiple of y i . Then x is not in the span of y − g , . . . , y g − since the y j ∈ I ( S ( K ) , µ ; s j )belong to different direct summands. Now, each map I ( V n , ¯ ω n ) : I ( S ( K ) , µ ) → I ( S n ( K ))has even degree, since it is a composition of the even-degree maps I ( W k , ω k ) for 1 ≤ k ≤ n .Thus, we have that I ( V n , ¯ ω n )( x ) ∈ I ( S n ( K ))for all n ≥
1, and since x is not in the span of y − g , . . . , y g − , Lemma 7.5 tells us that each I ( V n , ¯ ω n )( x ) must be nonzero. In particular, we have thatdim I ( S n ( K )) > n ≥
1, and so none of the S n ( K ) can be an instanton L-space. This is a contradiction,since if K is an instanton L-space knot then it has some positive integral L-space surgery,by Remark 2.20; so each I ( S ( K ) , µ ; s i ) must in fact be spanned by y i . (cid:3) Remark 7.6.
Nothing in the proof of Proposition 7.4 requires that we use the nontrivialbundle on S ( K ) specified by µ ; we only need this to conclude in Theorem 7.8 that K isfibered. If we use elements ˜ y i ∈ I ( S ( K ); s i ) instead and replace the bundle ν on X by˜ ν accordingly, none of the argument changes save perhaps some signs. Thus, it is also truethat if K is a nontrivial instanton L-space knot then each C -module I ( S ( K ); s i )is spanned by the single element ˜ y i = I ( X , ˜ ν ; t ,i )( ).The second main ingredient in the proof of Theorem 7.8 is the following improvement toProposition 7.4. Namely, Theorem 2.25 tells us that I ( S ( K ) , µ ; s i ) = 0 for all | i | ≥ g ( K ),but the proposition below says that we can sharpen this bound if we know that K has aninstanton L-space surgery. Proposition 7.7.
Suppose K ⊂ S as above is a nontrivial instanton L -space knot withsmooth slice genus g s ( K ) . Then I ( S ( K ) , µ ; s i ) = 0 for all i ≥ max( g s ( K ) , .Proof. Let g ′ s = max( g s ( K ) , . Then the cobordism X contains a smoothly embedded, closed surface S homologous to Σ ,built by gluing the core of the 2-handle used to form X to a genus- g ′ s surface in S × [0 , K × { } . We can thus improve the inequality (7.5) by replacing g = g (Σ )with g ′ s = g ( S ), so that I ( X , ν ; t ,i ) = 0 for all i ≥ g ′ s . This implies, in particular, that z ,i = I ( X , ν ; t ,i )( ) = 0for all i ≥ g ′ s . By Proposition 7.1, we have I ( W , ω )( y i ) = ±
12 ( z ,i + z ,i +1 ) , and so(7.18) I ( W , ω )( y i ) = 0for all i ≥ g ′ s .Now fix some i ≥ g ′ s , and suppose that y i = 0. Theorem 2.25 says that I ( S ( K ) , µ ; s i ) ∼ = I ( S ( K ) , µ ; s − i )as Z / Z -graded C -modules, so I ( S ( K ) , µ ; s − i ) is also nonzero; by Proposition 7.4, itmust be spanned by y − i , hence y − i = 0 as well. (We note that y i = y − i , because i ≥ g ′ s ≥ y i ∈ Im( I ( X , ν )) = C · I ( X , ν )( ) , NSTANTONS AND L-SPACE SURGERIES 51 so that y i = c (cid:16)P j y j (cid:17) for some c = 0, or, equivalently,( c − y i + X j = iy j =0 cy j = 0 . But the nonzero y j are linearly independent, as they belong to different direct summands y j ∈ I ( S ( K ) , µ ; s j ) , and the left hand side above is not identically zero because the term cy − i is nonzero. Thisis a contradiction, so y i = 0 after all. (cid:3) Finally, we apply the fact that framed instanton homology of 0-surgery detects fiberedness(Theorem 1.17) to obtain the promised restrictions on instanton L-space knots.
Theorem 7.8.
Suppose K ⊂ S is a nontrivial instanton L-space knot. Then K is fibered,and its Seifert genus is equal to its smooth slice genus.Proof. Theorem 1.17 and Proposition 7.4 combine to say that1 ≤ dim I ( S ( K ) , µ ; s g ( K ) − ) ≤ , respectively, and since the first inequality is then an equality we conclude that K must befibered. For the claim about the slice genus of K , we deduce from Proposition 7.7 that g ( K ) − < max( g s ( K ) , I ( S ( K ) , µ ; s g ( K ) − ) is nonzero, and hence that g s ( K ) ≤ g ( K ) ≤ max( g s ( K ) , . If g s ( K ) = 0 then we have g ( K ) ≤
1, but the only fibered knots of genus 1 are the trefoilsand the figure eight, which satisfy g s ( K ) = g ( K ) = 1. Otherwise max( g s ( K ) ,
1) = g s ( K )and so we must have g ( K ) = g s ( K ). (cid:3) The first L-space slope.
Supposing K is a nontrivial instanton L-space knot, ourgoal in this subsection is to determine in terms ofdim I ( S ( K ) , µ )which rational surgeries on K are instanton L-spaces (see Proposition 7.11). We begin withthe following lemma, which is a direct analogue of [KMOS07, Proposition 7.2]. Lemma 7.9.
Suppose K ⊂ S is a knot. For all k ≥ , let ( X ′ k +1 , η k +1 ) : S k +1 ( K ) → S be the cobordisms which induce the unlabeled maps in the exact triangles (7.2) (for k = 0 )and (7.3) (for k ≥ ). Then the composition I ( S k +1 ( K )) I ( X ′ k +1 ,η k ) −−−−−−−−→ I ( S ) I ( X k +1 ,ν k +1 ) −−−−−−−−−→ I ( S k +1 ( K )) is zero for all k ≥ . Proof.
This is essentially Lemma 4.13 and Remark 4.14 of [BS18], which in turn follows theproof of [KM95b, Proposition 6.5]. The composition is induced by a cobordism X : S k +1 ( K ) → S → S k +1 ( K ) , in which we attach a 0-framed 2-handle H µ to S k +1 ( K ) × [0 ,
1] along a meridian µ of K ×{ } and then attach a ( k + 1)-framed 2-handle H K to K in the resulting S . The cobordism X contains a smoothly embedded 2-sphere S of self-intersection zero, as the union of a cocoreof H µ and a core of H K .We wish to apply Proposition 6.7 to S , so we must construct a surface F with [ S ] · [ F ] = 0.The disjoint union of k + 1 parallel cores of H µ is bounded by a nullhomologous link in S k +1 ( K ) × { } , namely k + 1 disjoint copies of K , so let F be the union of these cores witha Seifert surface of ( k + 1) K and then we have [ S ] · [ F ] = k + 1 > (cid:3) Lemma 7.10. If I ( X k +1 , ν k +1 ) is injective for some k ≥ , then so is I ( X k , ν k ) .Proof. The injectivity of I ( X k +1 , ν k +1 ) implies by Lemma 7.9 that I ( X ′ k +1 , η k ) is zero;hence, I ( X k , ν k ) is injective by the exactness of either (7.2) or (7.3), depending on whether k = 0 or k ≥ (cid:3) We can now prove the main proposition of this subsection.
Proposition 7.11.
Let K ⊂ S be a nontrivial instanton L-space knot. Then the followingare equivalent for any rational r > :(1) r ≥ dim I ( S ( K ) , µ ) ,(2) S r ( K ) is an instanton L-space,(3) the map I ( X m , ν m ) : I ( S ) → I ( S m ( K )) is zero for m = ⌊ r ⌋ , where we interpret the codomain as I ( S ( K ) , µ ) when m = 0 .In particular, S g ( K ) − ( K ) is an instanton L-space.Proof. Let N ≥ S r ( K ) is an instanton L-space iff r ≥ N . Since S N − ( K )is not an instanton L-space, the map I ( X N − , ν N − ) appearing in the exact triangle · · · → I ( S ) I ( X N − ,ν N − ) −−−−−−−−−−→ I ( S N − ( K )) I ( W N ,ω N ) −−−−−−−−→ I ( S N ( K )) → . . . must be injective: otherwise it would be zero, leading todim I ( S N − ( K )) = dim I ( S N ( K )) − N − , which is a contradiction. (When N = 1, we interpret I ( S N − ( K )) here as I ( S ( K ) , µ );this has positive dimension because I ( S ( K ) , µ ; s g − )is nonzero by Theorem 1.17.) Lemma 7.10 now says that I ( X k , ν k ) is injective for 0 ≤ k ≤ N −
1. This shows that (3) = ⇒ (2), since if S r ( K ) is not an instanton L-space then0 ≤ m = ⌊ r ⌋ ≤ N − , and we have argued in this case that I ( X m , ν m ) is nontrivial. NSTANTONS AND L-SPACE SURGERIES 53
The argument above also shows, by the exactness of (7.2) and (7.3), that the maps I ( W k +1 , ω k +1 ) , ≤ k ≤ N − I ( S ) under the odd-degree map I ( X k , ν k ), and hence is supported entirely in odd grading. The composition I ( V N , ¯ ω N ) : I ( S ( K ) , µ ) → I ( S N ( K ))is thus surjective as well, with kernel an N -dimensional subspace of I ( S ( K ) , µ ). Since I ( V N , ¯ ω N ) has even degree, and I ( S N ( K )) = 0, we must have I ( S ( K ) , µ ) = ker I ( V N , ¯ ω N ) , and in particular N = dim I ( S ( K ) , µ ). This shows that (1) ⇐⇒ (2), since S r ( K ) is aninstanton L-space iff r ≥ N .We also know that the map I ( X N , ν N ) must be zero, because otherwise we deduce fromthe corresponding exact triangle thatdim I ( S N +1 ( K )) = dim I ( S N ( K )) − N − < N + 1 , which is impossible since I ( S N +1 ( K )) has Euler characteristic N + 1. Thus, Lemma 7.10implies that I ( X k , ν k ) = 0 for all k ≥ N . This shows that (2) = ⇒ (3), since if S r ( K ) isan instanton L-space then ⌊ r ⌋ ≥ N. For the final claim that S g − ( K ) is an instanton L-space, we know that each eigenspace I ( S ( K ) , µ ; s j ) = 0for | j | ≥ g ( K ), and each of the 2 g ( K ) − | j | ≤ g ( K ) − I ( S ( K ) , µ ) ≤ g ( K ) − , and we apply the equivalence (1) ⇐⇒ (2) to complete the proof of the claim. (cid:3) Recall from Proposition 7.4 that if K is a nontrivial instanton L-space knot then I ( S ( K ) , µ ; s i ) = C · y i for all i , where y i = I ( X , ν ; t ,i )( ) . Again, the nonzero y i are linearly independent because they belong to different eigenspaces,so Proposition 7.11 says that the smallest instanton L-space slope is precisely the numberof y i that are nonzero. Note that y i = 0 iff y − i = 0for each i , since the eigenspaces I ( S ( K ) , µ ; s ± i ) are isomorphic by Theorem 2.25 andare spanned by y ± i by Proposition 7.4. In particular, Theorem 1.17 implies that(7.19) y g ( K ) − = 0 and y − g ( K ) = 0in this case. Proposition 7.12.
Suppose K ⊂ S is a nontrivial knot, and that either S ( K ) or S ( K ) is an instanton L-space. Then K is the right-handed trefoil. Proof.
Let g = g ( K ), and suppose first that S ( K ) is an instanton L-space. Proposition 7.11implies that dim I ( S ( K ) , µ ) ≤ , and yet y g − and y − g are both nonzero as in (7.19). This is only possible if s g − = s − g , orequivalently if g = 1. Since K has genus 1 and it is fibered by Theorem 7.8, it can only be atrefoil or the figure-eight knot. But in [BS18, §
4] we noted that 1-surgery on the left-handedtrefoil and figure-eight are Seifert fibered and not the Poincar´e homology sphere, so theseare not instanton L-spaces by [BS18, Corollary 5.3]. Thus, K is the right-handed trefoil.Now suppose instead that S ( K ) is an instanton L-space, but S ( K ) is not. In this caseProposition 7.11 tells us that g ≥
2, and thatdim I ( S ( K ) , µ ) = 2 . The elements y g − and y − g are nonzero, and they lie in distinct eigenspaces since g ≥ I ( S ( K ) , µ ; s j ) must vanish, hence y j = 0 for 2 − g ≤ j ≤ g − . We now apply Proposition 7.1 to say that I ( W , ω ) g − X j =0 y − g +2 j = ǫ · g − X j =0 ( − − g +2 j ( z , − g +2 j + z , − g +2 j ) , or equivalently I ( W , ω ) (cid:0) y − g + y − g + y − g + · · · + y g − (cid:1) = ǫ − − g g − X j =2 − g z ,j = ± I ( X , ν )( ) . Now the left side vanishes since each of the y j appearing there is zero, so it follows that themap I ( X , ν ) is zero as well. But then Proposition 7.11 says that S ( K ) is an instantonL-space, and this is a contradiction. (cid:3) Corollary 7.13.
Suppose K ⊂ S is an instanton L-space knot of genus 1. Then K is theright-handed trefoil.Proof. Proposition 7.11 says that S ( K ) must be an instanton L-space, so we apply Propo-sition 7.12. (cid:3) Remark 7.14.
The proof of Proposition 7.12 also shows that if g = g ( K ) is odd and greaterthan 2, then S ( K ) is not an instanton L-space. Indeed, we know from the proposition inthis case that S ( K ) is not an instanton L-space, so if S ( K ) is, then Proposition 7.11 saysthat dim I ( S ( K ) , µ ) = 3 , which implies, by symmetry, that the nonzero y j are precisely y − g , y , and y g − . The factthat g is odd then means that the element y − g + y − g + y − g + · · · + y g − is again zero, which implies that I ( X , ν )( ) = 0, as in the proof of Proposition 7.12. Butthis implies by the proposition that S ( K ) is an instanton L-space, a contradiction. NSTANTONS AND L-SPACE SURGERIES 55 Instanton L-space knots are strongly quasipositive
The goal of this section is to prove the two theorems below, which, together with Theorem7.8, will complete the proof of Theorem 1.15.
Theorem 8.1.
Suppose K ⊂ S is a nontrivial instanton L-space knot. Then K is stronglyquasipositive. Theorem 8.2.
Suppose K ⊂ S is a nontrivial instanton L-space knot. Then S r ( K ) is aninstanton L-space for some rational r iff r ≥ g ( K ) − . We will begin with and spend the most time proving Theorem 8.1, proceeding as outlinedin the introduction. Namely, if K is an instanton L-space knot then it is fibered, by Theorem7.8. The corresponding fibration specifies an open book decomposition of S , and hence acontact structure ξ K on S by Thurston and Winkelnkemper’s construction [TW75]. Toprove that K is strongly quasipositive, it suffices to show that ξ K is the unique tight contactstructure ξ std on S , as recorded in the proposition below. Proposition 8.3.
Suppose K ⊂ S is a fibered knot. Then K is strongly quasipositive iff ξ K is tight, in which case the maximal self-linking number of K is sl ( K ) = 2 g ( K ) − .Proof. The relationship between strong quasipositivity and tightness was proved by Heddenin [Hed10, Proposition 2.1]. Hedden’s proof relies on the Giroux correspondence, but this isnot necessary: it was given an alternate proof and generalized to fibered knots in other 3-manifolds by Baker–Etnyre–van Horn-Morris [BEVHM12, Corollary 1.12]. The claim that if ξ K is tight then sl ( K ) = 2 g ( K ) − sl ( K ) ≤ g ( K ) − B of an open book supportingany contact structure ξ is transverse in ξ with self-linking number 2 g ( B ) − (cid:3) We will study ξ K via cabling. Let K p,q denote the ( p, q )-cable of K , where p and q arerelatively prime and q ≥ ∂N ( K ) ⊂ S representingthe class µ p λ q , for µ the meridian of K and λ the longitude). It is well-known that if K isfibered then so is K p,q , in which case we can talk about the corresponding contact structure ξ K p,q . For positive cables, this contact structure agrees with ξ K , as below. Proposition 8.4.
Suppose K ⊂ S is a fibered knot. Then ξ K ∼ = ξ K p,q for p and q positive.Proof. This was proven by Hedden in [Hed08, Theorem 1.2] using the Giroux correspon-dence. An alternate proof which does not rely on this correspondence was given by Baader–Ishikawa in [BI09, § (cid:3) We will also use the following, which states that the ( p, q )-cable of an instanton L-spaceknot is itself an instanton L-space knot for pq sufficiently large. Lemma 8.5.
Suppose K ⊂ S is a nontrivial knot, and fix positive, coprime integers p and q with q ≥ and pq > g ( K ) − . Then K is an instanton L-space knot iff K p,q is.Proof. We note that pq − q > p − q ≥ g ( K ) −
1, and also that g ( K p,q ) ≤ g ( T p,q ) + q · g ( K ) , which after some rearranging yields pq − ≥ g ( K p,q ) − q (cid:18) p − q − (2 g ( K ) − (cid:19) ≥ g ( K p,q ) − . Thus, Proposition 7.11 implies that K is an instanton L-space knot iff pq − q -surgery on K is an instanton L-space, and likewise that K p,q is an instanton L-space knot iff S pq − ( K p,q )is an instanton L-space. The lemma now follows immediately from the relation S pq − ( K p,q ) ∼ = S pq − /q ( K ) , which was originally proved by Gordon [Gor83, Corollary 7.3]. (cid:3) Our strategy in proving Theorem 8.1 is roughly as follows: if K is a nontrivial instantonL-space knot then so is a positive cable K p,q with pq sufficiently large, by Lemma 8.5. Wewill use this together with the fact that K p,q is a Murasugi sum of the form K ,q ∗ T p,q toprove that the contact structure ξ K ,q is tight (for q = 2, though it holds more generally).This will imply that ξ K is tight and hence that K is strongly quasipositive, by Propositions8.4 and 8.3. Our proof that ξ K ,q is tight makes use of a variation of our instanton contactclass from [BS16], developed in the next subsection.8.1. Open books and framed instanton homology.
In this subsection, we briefly de-scribe a variant of the contact classΘ( ξ ) ∈ SHI ( − M, − Γ)which we constructed for sutured contact manifolds in [BS16]. Here, we specialize ourconstruction to closed contact 3-manifolds, as in [BS18], and define this variant as a subspaceof framed instanton homology, rather than an element of sutured instanton homology upto rescaling. We explain the reason for this in Remark 8.10.The following definitions are all taken from [BS18, § Definition 8.6. An abstract open book is a triple ( S, h, c ) consisting of a surface S withnonempty boundary, a diffeomorphism h : S → S such that h | ∂S = id ∂S , and a collectionof disjoint, properly embedded arcs c = { c , . . . , c b ( S ) } ⊂ S such that S r c deformationretracts onto a point. Definition 8.7.
An abstract open book (
S, h, c ) determines a contact 3-manifold M ( S, h, c )with convex boundary S as follows. We take a contact handlebody H ( S ) ∼ = S × [ − ,
1] witha tight, [ − , ∂H ( S ) is a convex surface identified with the double of S and its dividing set is Γ = ∂S .We then attach contact 2-handles to H ( S ) along the collection γ ( h, c ) of curves γ i = (cid:0) c i × { } (cid:1) ∪ (cid:0) ∂c i × [ − , (cid:1) ∪ (cid:0) h ( c i ) × {− } (cid:1) for 1 ≤ i ≤ b ( S ), and denote the result by M ( S, h, c ). Definition 8.8. An open book decomposition of a based contact 3-manifold ( Y, ξ, p ) is atuple B = ( S, h, c , f )consisting of an abstract open book ( S, h, c ) and a contactomorphism f : M ( S, h, c ) → ( Y ( p ) , ξ | Y ( p ) ) , in which Y ( p ) is the complement of a Darboux ball around p . NSTANTONS AND L-SPACE SURGERIES 57
Let B = ( S, h, c , f ) be an open book decomposition of ( Y, ξ, p ). We choose a compact,oriented surface T and an identification ∂T ∼ = ∂S , let R = S ∪ T , and extend h to R so that h | T = id T . We also form the mapping torus R × φ S = R × [ − , x, ∼ ( φ ( x ) , − φ : R → R . Then S × [ − ,
1] is naturally a submanifold of R × φ S ,so we can view the collection of curves γ ( h, c ) of Definition 8.7 as living inside R × φ S . Weproved the following in [BS18, Proposition 2.16]. Proposition 8.9.
Performing ∂ ( S × [ − , -framed surgery on each γ i ∈ γ ( h, c ) inside R × φ S produces a manifold which is canonically diffeomorphic to M ( S, h, c ) ∪ ∂ (( R × h − ◦ φ S ) r B ) up to isotopy. Using the diffeomorphism f : M ( S, h, c ) → Y ( p ), we define a cobordism V B ,φ : R × φ S → Y R × h − ◦ φ S )by attaching 2-handles to ( R × φ S ) × [0 ,
1] along each of the ∂ ( S × [ − , γ i ∈ γ ( h, c ). Finally, we fix a closed curve α ∈ R × φ S , disjoint from a neighborhood of S × [ − , α ∩ ( R × [ − , { t } × [ − , t ∈ int( T ). We also use α to denote its image in Y R × h − ◦ φ S ).We now recall from Proposition 4.4 that(8.1) I ( − R × φ S , α |− R ) ∼ = I ( − R × φ S , α |− R ) ∼ = C , where we are using the notation from the end of § ( B , R, φ ) ⊂ I ( − Y − R × h − ◦ φ S ) , α |− R )to be the image of the map induced by the cobordism ( − V B ,φ , α × [0 , − R , which we denote as in § I ( − V B ,φ , α × [0 , |− R ) : I ( − R × φ S , α |− R ) → I ( − Y − R × h − ◦ φ S ) , α |− R ) . Since I ( − V B ,φ , α × [0 , Z / Z -grading, by Proposition2.21, the subspace Θ ( B , R, φ ) is the direct sum of two subspaces of dimension at most 1,one in even grading and one in odd grading. Remark 8.10.
The reason we define Θ ( B , R, φ ) as the image of I ( − R × φ S , α |− R )under the map (8.2) rather than as the image of some element under this map (whichwould be more in line with the definition of the contact invariant in [BS16]), is that thelatter requires choosing an element of I ( − R × φ S , α |− R ). There are two natural choices,in light of (8.1), corresponding to the generators in even and odd gradings, but it is notclear which one is preferred. Moreover, it is not clear whether the excision isomorphismsrelating the Floer groups associated to different closures of S × [ − ,
1] (corresponding todifferent choices of T and φ ) preserve the Z / Z -grading, which means that it is not clearwhether one can actually define an invariant contact element in this setting by choosing theeven or odd generator. Definition 8.11. A positive stabilization of the abstract open book ( S, h, c ) is an openbook ( S ′ , h ′ = D β ◦ h, c ′ = c ∪ { c } ) , where S ′ is formed by attaching a 1-handle H to S with cocore c , and D β is a right-handedDehn twist about some closed curve β ⊂ S ′ which intersects c transversely in a single point.There is then a canonical contactomorphism q : M ( S ′ , h ′ , c ′ ) ∼ −→ M ( S, h, c )up to isotopy, and thus an open book decomposition B = ( S, h, c , f ) of ( Y, ξ, p ) can bepositively stabilized to an open book decomposition B ′ = ( S ′ , h ′ , c ′ , f ′ = f ◦ q ) . See [BS16, § Proposition 8.12.
Let B ′ = ( S ′ , h ′ = D β ◦ h, c ′ , f ′ ) be a positive stabilization of B =( S, h, c , f ) , and embed S ′ in a closed surface R as above. Then Θ ( B ′ , R, φ ) = Θ ( B , R, D − β ◦ φ ) for any diffeomorphism φ : R → R .Proof. The proof is identical to that of [BS16, Proposition 4.5]. The point is that we cancompute Θ ( B ′ , R, φ ) as the image of the composition of a certain cobordism mapΨ φ,β : I ( − R × φ S , α |− R ) → I ( − R × D − β ◦ φ S , α |− R )with the cobordism map I ( − V B ,D − β ◦ φ , α × [0 , |− R ) whose image is the subspaceΘ ( B , R, D − β ◦ φ ) ⊂ I ( − Y − R × h − ◦ ( D − β ◦ φ ) S ) , α |− R ) . (Note that h − ◦ ( D − β ◦ φ ) = ( D β ◦ h ) − ◦ φ = ( h ′ ) − ◦ φ .) Now, the map Ψ φ,β is inducedby the cobordism obtained by attaching a 2-handle along a copy of β , viewed as a curve insome fiber − R . It thus fits into a surgery exact triangle in which the remaining entry is zerobecause the surgery compresses − R , making it homologous to a surface of strictly lowergenus. We conclude that Ψ φ,β is an isomorphism when restricted to the top eigenspaces of µ ( − R ), from which the proposition follows; see [BS16] for more details. (cid:3) We wish to use the subspaces Θ ( B , R, φ ) defined above to obstruct overtwistedness (seeCorollary 8.15). The following proposition is the technical key behind this. Proposition 8.13.
Let B = ( S, h, c , f ) be an open book decomposition of a based con-tact manifold ( Y, ξ, p ) . Suppose that some page contains a nonseparating, nullhomologousLegendrian knot Λ for which the contact framing of Λ agrees with its page framing, andtb (Λ) ≥ g (Λ) > . Let R be a surface containing S as a subsurface, as above. Then Θ ( B , R, φ ) = { } for alldiffeomorphisms φ : R → R .Proof. This is proved by combining the arguments of [BS16, Proposition 4.6] and [BS16,Theorem 4.10], without any substantial changes. We outline these arguments below for thereader’s benefit.
NSTANTONS AND L-SPACE SURGERIES 59
First, we form ( Y − , ξ − ) by Legendrian surgery along Λ, and let the cobordism X : Y → Y − denote its trace. Then ( Y − , ξ − ) is supported by an open book B − = ( S, D Λ ◦ h, c , f − ) , and if we let Λ − be the image of a Legendrian push-off of Λ in ( Y − , ξ − ), then we can recover( Y, ξ ) by performing contact (+1)-surgery on Λ − . The trace of this contact (+1)-surgeryis diffeomorphic to the reversed cobordism X † : − Y − → − Y , and it follows exactly as in[BS16, Proposition 4.6] that the map I ( − Y − − R × h − ◦ φ S ) , α |− R ) → I ( − Y − R × h − ◦ φ S ) , α |− R )induced by the cobordism X † ,⊲⊳ := X † ⊲⊳ (cid:0) ( − R × h − ◦ φ S ) × [0 , (cid:1) sends Θ ( B − , R, D Λ ◦ φ ) to Θ ( B , R, φ ).Now we let Σ ⊂ Y be a Seifert surface for Λ of minimal genus g = g (Λ), and cap it offinside X to form a closed surface ˆΣ of genus g and self-intersection tb (Λ) − ≥ g −
1. Thissurface persists inside X † ,⊲⊳ , and since we have ˆΣ · ˆΣ > g ( ˆΣ) − I ( X † ,⊲⊳ , α × [0 , |− R ) = 0 . But Θ ( B , R, φ ) lies in the image of this map, so it must be zero. (cid:3) Remark 8.14.
Strictly speaking, the adjunction inequality as stated in Proposition 6.7requires that b ( X † ,⊲⊳ ) = 0, though we can appeal more generally to [KM95b] here. Alter-natively, if Y is a rational homology sphere then b ( X † ) = 0, and one can argue that theconnected sum of either − Y or − Y − with ( − R × h − ◦ φ S ) T has “strong simple type” inthe sense of Mu˜noz [Mu˜n00], specifically when restricting to the (2 g ( R ) − , µ ( − R ) , µ (pt), and then repeat verbatim the proof of the structure theorem which led toProposition 6.7. This case will suffice for our purposes, in which Y = S . Corollary 8.15.
Suppose ξ is an overtwisted contact structure on a based manifold ( Y, p ) .Then there is a supporting open book decomposition of ( Y, ξ, p ) , B ot = ( S ot , h ot , c ot , f ot ) , such that Θ ( B ot , R, φ ) = { } for any R containing S ot as a subsurface, and all diffeomorphisms φ : R → R .Proof. That ξ is overtwisted implies that we can find a Legendrian right-handed trefoil Λcontained in a ball in Y with tb (Λ) = 2 (see [BS16, Proof of Theorem 4.1]), violating theThurston–Bennequin inequality. There exists a supporting open book decomposition B ot = ( S ot , h ot , c ot , f ot )in which Λ lies in a page as a nonseparating curve, with contact framing equal to its pageframing. The corollary then follows from Proposition 8.13. (cid:3) Open books from cables of L-space knots.
For the rest of this subsection, let usfix a nontrivial instanton L-space knot K and positive, coprime integers p and q with q ≥ pq > g ( K ) −
1, so that K p,q is also an instanton L-space knot, by Lemma 8.5, and ξ K p,q ∼ = ξ K ,q ∼ = ξ K , by Proposition 8.4. Let B p,q = ( S p,q , h p,q , c p,q , f p,q )be an open book decomposition of ( S , ξ K p,q ) with binding K p,q which encodes the fibrationassociated to the fibered knot K p,q . That is, S p,q is a fiber surface of K p,q , h p,q : S p,q −→ S p,q the monodromy of the fibration, and c p,q is some basis of arcs for S p,q . We will prove thatΘ ( B ,q , R, h ) = { } for an appropriate choice of R ⊃ S ,q and h (see Proposition 8.17). We will then use this incombination with Proposition 8.12 and Corollary 8.15 to conclude that ξ K ,q ∼ = ξ K is tight(though will only carry this out explicitly for q = 2), proving Theorem 8.1.The reason we consider cables is that they can be deplumbed. More precisely, Neumann–Rudolph proved in [NR87, § K p,q is a Murasugi sum of K ,q with thetorus knot T p,q ; see [NR87, Figure 4.2] or [BEVHM12, Figure 1]. We can express this as aMurasugi sum of abstract open books, by( S p,q , h p,q ) ∼ = ( S ,q , h ,q ) ∗ (Σ p,q , φ p,q ) , where (Σ p,q , φ p,q ) is an open book for the fibration associated with T p,q . But (Σ p,q , φ p,q ) canitself be constructed by plumbing together 1 − χ (Σ p,q ) = 2 g ( T p,q ) positive Hopf bands; see,e.g., [OS04, § S p,q , h p,q ) is obtained from the open book(8.3) ( S p,q , h ,q ) := ( S ,q , h ,q ) ∗ (Σ p,q , id)by adding a right-handed Dehn twist to the monodromy along the core of each Hopf bandused to construct Σ p,q . Note that the open book ( S p,q , h ,q ) defined in (8.3) supports thecontact manifold (cid:16) S (cid:16) g ( T p,q ) ( S × S ) (cid:17) , ξ K ξ std (cid:17) , since ( S ,q , h ,q ) supports the contact structure ξ K ,q ∼ = ξ K on S and any open book of theform ( S, id) supports the unique tight contact structure ξ std on − χ ( S ) ( S × S ) . For notational simplicity, let Y := g ( T p,q ) ( S × S ) , which we identify with the manifold supported by the open book ( S p,q , h ,q ), and let J ⊂ Y denote the binding of this open book. Then g ( J ) = g ( K p,q ), since J and K p,q both boundminimal genus Seifert surfaces Σ ⊂ Y and Σ ′ ⊂ S which are identified with S p,q .Let ( X , ν ) : ( S , → ( S ( K p,q ) , µ )be the cobordism given by the trace of 0-surgery on K p,q , where µ is a meridian of K p,q and ν is the surface used in §
7. As mentioned above, the monodromy h p,q is obtained from h ,q by adding right-handed Dehn twists along curves(8.4) c , . . . , c g ( T p,q )NSTANTONS AND L-SPACE SURGERIES 61 in Σ p,q ⊂ S p,q corresponding to cores in a plumbing description of Σ p,q . It follows that S isobtained from Y by performing − c i on g ( T p,q ) parallel pagesof the open book decomposition ( S p,q , h ,q ) of Y , with respect to the page framings of thesecurves. The cobordism ( X , ν ) above therefore fits into a commutative diagram(8.5) ( Y, ( X,ν ) / / (cid:15) (cid:15) ( Y ( J ) , µ ) (cid:15) (cid:15) ( S , ( X ,ν ) / / ( S ( K p,q ) , µ ) , with the cobordism ( X, ν ) : ( Y, → ( Y ( J ) , µ )given by the trace of 0-surgery on J , where µ is a meridian of J , and ν is the correspondingsurface. The vertical arrows in this diagram correspond to the cobordisms obtained as thetrace of − c i , viewed as curves in Y and Y ( J ). Commutativity followsfrom the fact that these curves are disjoint from J and µ .The first step toward proving that ξ K ,q is tight is the following lemma. Lemma 8.16.
Let ˆΣ ′ ⊂ Y ( J ) denote the closed surface obtained by capping off the Seifertsurface Σ ′ ∼ = S p,q for J , and let ( X † , ν † ) : ( − Y ( J ) , µ ) → ( − Y, be the cobordism obtained by turning ( X, ν ) upside-down. Then the induced map (8.6) I ( X † , ν † |− ˆΣ ′ ) : I ( − Y ( J ) , µ |− ˆΣ ′ ) → I ( − Y ) is nonzero.Proof. The commutative diagram (8.5) gives rise to the commutative diagram(8.7) I ( − S ( K p,q ) , µ |− ˆΣ) I ( X † ,ν † |− ˆΣ) / / (cid:15) (cid:15) I ( − S ) (cid:15) (cid:15) I ( − Y ( J ) , µ |− ˆΣ ′ ) I ( X † ,ν † |− ˆΣ ′ ) / / I ( − Y ) , whose arrows are given by the maps induced by the cobordisms in (8.5) turned upside-downand restricted to the top eigenspaces of µ ( ˆΣ ′ ) and µ ( ˆΣ), where ˆΣ is the closed surface in S ( K p,q ) obtained by capping off the Seifert surface Σ ∼ = S p,q for K p,q .We claim that the rightmost vertical map I ( − S ) → I ( − Y ) = I ( − g ( T p,q ) ( S × S ))is injective. Indeed, each curve c i is dual to a 2-sphere in a unique S × S summand of Y = g ( T p,q ) ( S × S ), which at the level of open books comes from plumbing an annuluswith trivial monodromy onto ( S ,q , h ,q ). It follows that the map I ( − S ) → I ( − Y ) isinduced by 0-surgery on a 2 g ( T p,q )-component unlink, and so an argument using the surgeryexact triangle (2.6) exactly as in [Sca15, § I ( X , ν ; t , − g ( K p,q ) ) : I ( S ) → I ( S ( K p,q ) , µ ; s − g ( K p,q ) ) in the notation of §
7, where in particular s − g ( K p,q ) : H ( S ( K p,q )) → Z is the homomorphism defined by s − g ( K p,q ) ([ ˆΣ]) = 2 − g ( K p,q ) = 2 − g ( ˆΣ) . Since K p,q is an instanton L-space knot, Theorem 1.17 and Proposition 7.4 (together withthe conjugation symmetry of Theorem 2.25) imply that the map (8.8) is injective, withimage spanned by the nonzero element y − g ( K p,q ) ∈ I ( S ( K p,q ) , µ ; s − g ( K p,q ) ) . Thus, the dual I ( X † , ν † |− ˆΣ) of this map is also nonzero.Since the diagram (8.7) is commutative and the composition of top and right maps isnonzero, we conclude that the same must be true for the composition of the left and bottommaps. In particular, the bottom map I ( X † , ν † |− ˆΣ ′ ) : I ( − Y ( J ) , µ |− ˆΣ ′ ) → I ( − Y )must also be nonzero, as claimed. (cid:3) To prove that the subspace Θ ( B ,q , R, h ) is nontrivial for some R and h , we relate thecobordism map in Lemma 8.16 with the map which defines this subspace. First note thatthe surface S p,q = S ,q ∗ Σ p,q can also be expressed as S p,q = S ,q ∪ T, where T is a surface of genus g ( T p,q ) and one boundary component, along which it is gluedto ∂S ,q . The 0-surgery Y ( J ) is then given by the mapping torus, Y ( J ) = ( S ,q ∪ T ) × h S , where the map h : S ,q ∪ T → S ,q ∪ T is equal to h ,q on S ,q and the identity on T . Let us define R := S ,q ∪ T, and observe that under the identification Y ( J ) = R × h S , the capped-off Seifert surface ˆΣ ′ in Lemma 8.16 is identified with a copy of the fiber R . In particular, g ( R ) = g ( K p,q ) = g ( ˆΣ ′ ).Recall that the subspaceΘ ( B ,q , R, h ) ⊂ I ( − S − R × S ) , α |− R )is the image of the map I ( − V B ,q ,h , α × [0 , |− R ) : I ( − R × h S , α |− R ) → I ( − S − R × S ) , α |− R )induced by the cobordism V B ,q ,h : R × h S → S R × S ) , obtained by attaching ∂ ( S ,q × [ − , R × h S ) × [0 ,
1] along each ofthe 2 g ( K ,q ) curves γ j × { } for γ j ∈ γ ( h ,q , c ,q ) , where c ,q is a basis of arcs for S ,q and α is the meridian µ of J in − Y ( J ) = − R × h S . NSTANTONS AND L-SPACE SURGERIES 63
Proposition 8.17.
The cobordism map I ( − V B ,q ,h , α × [0 , |− R ) : I ( − R × h S , α |− R ) → I ( − S − R × S ) , α |− R ) above is nonzero. In particular, Θ ( B ,q , R, h ) = { } . Proof.
We prove that the cobordism( X † , ν † ) : ( − Y ( J ) , µ ) → ( − Y, − V B ,q ,h , α × [0 , X † , ν † ) : ( − R × h S , α ) → ( − g ( T p,q ) ( S × S ) , , per the discussion preceding the proposition. For the claimed factorization, first note that X † is the cobordism given by the trace of 0-surgery on a section { pt } × S of − T × S ⊂ − R × h S . In the surgered manifold − g ( T p,q ) ( S × S ), the induced curves γ j ∈ γ ( h ,q , c ,q ) are eachjust unknots with page framing equal to their 0-framing. Let W : − g ( T p,q ) ( S × S ) → − g ( K p,q ) ( S × S )be the 2-handle cobordism given by the trace of 0-surgeries on these unknots, and let Z : − g ( K p,q ) ( S × S ) → − g ( T p,q ) ( S × S )be the 3-handle cobordism cancelling these 2-handles. Then the composition − R × h S X † −−→ − g ( T p,q ) ( S × S ) W −→ − g ( K p,q ) ( S × S ) Z −→ − g ( T p,q ) ( S × S )is simply X † .We can now change the order of the first two sets of 2-handle attachments since they areperformed along the disjoint curves { pt }× S and γ , . . . , γ g ( K ,q ) ∈ γ ( h ,q , c ,q ). Performingthe latter set of 2-handle attachments first results in the cobordism( − V B ,q ,h , α × [0 , − R × h S , α ) → ( − R × S , α ) . This shows that X † can also be expressed as a composition of the form − R × h S − V B ,q,h −−−−−→ − R × S W ′ −−→ − g ( T p,q ) ( S × S )for some cobordism ( W ′ , ν ′ ) : ( − R × S , α ) → ( − g ( T p,q ) ( S × S ) , I ( X † , ν † |− R ) = I ( W ′ , ν ′ |− R ) ◦ I ( − V B ,q ,h , α × [0 , |− R ) . Lemma 8.16 says that the map I ( X † , ν † |− R ) is nontrivial (recall that ˆΣ is identified witha copy of the fiber R here); hence, so is the map I ( − V B ,q ,h , α × [0 , |− R ) . Since its image is by definition Θ ( B ,q , R, h ), this subspace is nonzero. (cid:3) Remark 8.18.
As a special case of the constructions above, let q = 2 and p = 2 k + 1 with k ≥ g ( K ) − pq ≥ g ( K ) − g ( R ) = g ( K , ) + g ( T k +1 , ) = 2 g ( K ) + k. Thus by varying k , we can arrange for g ( R ) to be any integer which is at least 4 g ( K ) − h : R → R of Proposition 8.17 is the identity on Σ k +1 , ⊂ R , and itsrestriction h | S , ⊂ R = h , does not depend on the choice of k .We may now prove Theorem 8.1. Proof of Theorem 8.1.
According to Propositions 8.3 and 8.4, it suffices to show that ξ K , is not overtwisted. Supposing otherwise, there exists by Corollary 8.15 a supporting openbook decomposition for ( S , ξ K , ) given by B ot = ( S ot , h ot , c ot , f ot ) , such that(8.9) Θ ( B ot , R, φ ) = { } for any R containing S ot as a subsurface, and all diffeomorphisms φ : R → R . The Girouxcorrespondence [Gir02] asserts that there is an open book decomposition B + = ( S + , h + , c + , f + )which is simultaneously a positive stabilization of both B ot and the open book decomposition B , = ( S , , h , , c , , f , )described at the beginning of this subsection.Let us now embed S + in a closed, connected surface R of genus at least 4 g ( K ) − R r S + of the image is connected with positive genus;this induces embeddings S ot , S , ֒ → R satisfying the same conditions. The vanishing (8.9)together with Proposition 8.12 then implies thatΘ ( B + , R, φ ) = { } for all φ as well. On the other hand, Proposition 8.17 says thatΘ ( B , R, h ) = { } , where h | S , = h , and h | R r S , = id, so another application of Proposition 8.12 says thatΘ ( B + , R, h ′ ) = { } for some corresponding h ′ , which is a contradiction. (cid:3) Instanton L-space slopes.
Suppose K ⊂ S is a nontrivial instanton L-space knot.We proved in [BS18, Theorem 4.20] that there exists a positive integer N such that S r ( K )is an instanton L-space iff r ∈ [ N, ∞ ) ∩ Q , and we showed in Proposition 7.11 that N ≤ g ( K ) − . Furthermore, it follows from Theorem 8.1 and Proposition 8.3 that(8.10) sl ( K ) = 2 g ( K ) − . NSTANTONS AND L-SPACE SURGERIES 65
Lidman, Pinz´on-Caicedo, and Scaduto prove as part of their forthcoming work in [LPCS19]that K being a nontrivial instanton L-space knot together with (8.10) implies that(8.11) N ≥ g ( K ) − . Thus, N = 2 g ( K ) −
1, proving Theorem 8.2. The argument for the bound (8.11), which isonly part of the main work in [LPCS19], is simple enough that we reproduce it below forcompleteness.
Proof of Theorem 8.2.
As discussed above, all that remains is to prove the inequality (8.11).Note by Proposition 7.11 that this is equivalent to(8.12) dim I ( S ( K ) , µ ) ≥ g ( K ) − . Since χ ( I ( S ( K ) , µ )) = 0 by Proposition 2.18, we have thatdim I ( S ( K ) , µ ) = dim I ( S ( K ) , µ ) , which implies that (8.12) is equivalent to(8.13) dim I ( S ( K ) , µ ) ≥ g ( K ) − . The latter is what we will prove below.Let g = g ( K ). Since sl ( K ) = 2 g −
1, we can find a Legendrian representative Λ of K inthe standard contact S with classical invariants( tb (Λ) , r (Λ)) = ( τ , r ) , τ − r = 2 g − sl transverse representative of K will do). For n ≥ − τ , we can positively stabilize this Legendrian k times and negatively stabilize it τ + n − − k times to get a Legendrian representative with( tb , r ) = (1 − n, − g − n + 2 k ) , ≤ k ≤ τ + n − . For odd n ≫
0, these values of r include every positive odd number between 1 and n +2 g − n , we perform Legendrian surgery on these knots Λ i with( tb (Λ i ) , r (Λ i )) = (1 − n, i − , ≤ i ≤ n + 2 g − , to get contact structures ξ , . . . , ξ ( n +2 g − / on Y := S − n ( K ) . Let W := X − n ( K )be the trace of this − n -surgery, and ˆΣ ⊂ W the union of a Seifert surface for K with thecore of the 2-handle. Then each ξ i admits a Stein filling ( W, J i ) with h c ( J i ) , [ ˆΣ] i = r (Λ i ) = 2 i − . We can also take contact structures¯ ξ i = T ( Y ) ∩ ¯ J i T ( Y ) , ≤ i ≤ n + 2 g − , which are filled by W with the conjugate Stein structure ¯ J i for each i . These satisfy h c ( ¯ J i ) , [ ˆΣ] i = − (2 i − , so we have exhibited n + 2 g − J , J , . . . , J ( n +2 g − / , ¯ J , ¯ J , ¯ J ( n +2 g − / on W which are all distinguished by their first Chern classes. This implies by [BS18,Theorem 1.6] that the associated contact invariants(8.14) Θ( ξ ) , . . . , Θ( ξ ( n +2 g − / ) , Θ( ¯ ξ ) , . . . , Θ( ¯ ξ ( n +2 g − / )are linearly independent as elements of the sutured instanton homology of the complementof a ball in − Y (with suture consisting of one circle). Concretely, we showed in [BS18] thatthese contact invariants can all be thought of simultaneously as elements of I ∗ ( − Y R × S ) |− R ) α ∪ η for some fixed closed surface R , where α = { pt } × S for some pt ∈ R , and η is a homolog-ically essential curve in some copy of the fiber R . By removing a 4-ball from W † , we mayview it as a cobordism W † : − Y → − S . Let W † ,⊲⊳ be the cobordism W † ,⊲⊳ := W † ⊲⊳ (cid:0) ( − R × S ) × [0 , (cid:1) : − Y R × S ) → − S R × S ) . The functoriality of our contact invariants under maps induced by Stein cobordisms implies[BS16, BS18] that the map induced by ( W † ,⊲⊳ , ( α ∪ η ) × [0 , I ∗ ( − Y R × S ) |− R ) α ∪ η → I ∗ ( − S R × S ) |− R ) α ∪ η ∼ = C , sends each of the contact classes (8.14) to a generator. This implies that these classes allhave the same mod 2 grading, since cobordism maps are homogeneous.Now, Kronheimer and Mrowka showed in [KM10] that I ( − Y ) ∼ = I ∗ ( − Y R × S ) |− R ) α ∪ η as part of their proof that sutured instanton homology is independent of the closure. Thisisomorphism is a composition of excision isomorphisms, and is therefore homogeneous withrespect to the mod 2 grading. So, it identifies the contact classes in (8.14) with n + 2 g − x , . . . , x ( n +2 g − / , ¯ x , . . . , ¯ x ( n +2 g − / ∈ I ( − Y ) := I ( − S − n ( K ))which all have the same mod 2 grading. The fact that χ ( I ( − S − n ( K ))) = n then forces there to be at least 2 g − I ( − S − n ( K ))in the other mod 2 grading. In conclusion, we have shown thatdim I ( − S − n ( K )) ≥ n + 4 g − . By an easy application of the surgery exact triangles · · · → I ( − S ) → I ( − S ( K ) , µ ) → I ( − S − ( K )) → . . . and · · · → I ( − S ) → I ( − S − k ( K )) → I ( − S − ( k +1) ( K )) → . . . , we conclude that dim I ( − S ( K ) , µ ) = dim I ( S ( K ) , µ ) ≥ g − , NSTANTONS AND L-SPACE SURGERIES 67 as desired. (cid:3)
Proof of Theorem 1.15.
This is simply a combination of Theorems 7.8, 8.1, and 8.2. (cid:3)
Observe that Theorem 8.2 and Proposition 7.11 together imply that if K is a nontrivialinstanton L-space knot, then dim I ( S ( K ) , µ ) = 2 g ( K ) − . We know from Proposition 7.4 that each I ( S ( K ) , µ ; s i ) is spanned by the element y i = I ( X , ν ; t ,i ) for | i | ≤ g ( K ) −
1, and is zero for i outside this range. The y i span a space ofdimension equal to the number of nonzero y i , since these are linearly independent. Thus wecan characterize instanton L-space knots in terms of 2-handle cobordism maps as follows. Corollary 8.19.
A nontrivial knot K ⊂ S is an instanton L-space knot iff the cobordismmaps I ( X , ν ; t ,i ) : I ( S ) → I ( S ( K ) , µ ; s i ) are isomorphisms for all i in the range − g ( K ) ≤ i ≤ g ( K ) − . (cid:3) As in Remark 7.6, this corollary does not require using a nontrivial bundle on S ( K ) asspecified by µ ; we can also conclude that K is an instanton L-space knot iff the maps I ( X ; t ,i ) : I ( S ) → I ( S ( K ); s i )are isomorphisms for 1 − g ( K ) ≤ i ≤ g ( K ) − SU (2) representation varieties of Dehn surgeries In this section, we apply our results about instanton L-space knots and their L-spacesurgeries to questions about the SU (2) representation varieties of Dehn surgeries on knots in S . As explained in the introduction, our main tool is the following [BS18, Corollary 4.8]; wehave already used this (in the introduction) to show that Theorem 1.15 implies Theorems 1.5and 1.10. Proposition 9.1.
Suppose K ⊂ S is a nontrivial knot and r = mn > is a rational numbersuch that ∆ K ( ζ ) = 0 for any m th root of unity ζ . If S r ( K ) is SU (2) -abelian then S r ( K ) is an instanton L-space. Corollary 9.2.
Suppose K ⊂ S is a nontrivial knot and r = mn > is a rational number inwhich m is a prime power. If S r ( K ) is SU (2) -abelian then S r ( K ) is an instanton L-space.Proof. Write m = p e for some prime p and e ≥
1. If S r ( K ) is not an instanton L-spacethen Proposition 9.1 says that there is some p e th root of unity ζ for which ∆ K ( ζ ) = 0. Inother words, if Φ k ( t ) is the cyclotomic polynomial of order k , then Φ p e ( t ) divides ∆ K ( t )as elements of Z [ t, t − ]. Setting t = 1, it follows that Φ p e (1) = p divides ∆ K (1) = 1, acontradiction. (cid:3) Remark 9.3.
The set of rational numbers r = mn > m a prime power is dense in[0 , ∞ ). Indeed, given any rational s = ab >
0, the rational numbers s ± kb = ka ± kb , k ≥ are also positive and approach s as k goes to infinity, and for either choice of sign there areinfinitely primes congruent to ± a ), hence infinitely many k such that the numerator ka ± Proposition 9.4.
Suppose K ⊂ S is a nontrivial knot and r = mn ∈ (0 , is a rationalnumber with m a prime power. Then S r ( K ) is not SU (2) -abelian.Proof. Suppose for a contradiction that S r ( K ) is SU (2)-abelian. Then S r ( K ) is an instan-ton L-space, by Corollary 9.2. It follows that S ⌊ r ⌋ ( K ) is an instanton L-space as well, by[BS18, Theorem 4.20]. This is impossible by definition for r ∈ (0 , r ∈ [1 , ⌊ r ⌋ is equal to 1 or 2. Proposition 7.12 then tells us that K isthe right-handed trefoil T , . So to finish the proof, we need only show that S r ( T , ) is not SU (2)-abelian for any r ∈ [1 , r is not an integer. Indeed, when r = 1 we know that the fundamentalgroup of S ( T , ) = − Σ(2 , ,
5) is already a non-abelian subgroup of SU (2). In the case r =2, it is not hard to show that the trefoil group admits an irreducible SU (2) representation ρ satisfying ρ ( µ ) = e πi/ and ρ ( λ ) = e πi/ as unit quaternions, and then ρ ( µ λ ) = 1, so ρ descends to an irreducible representation π ( S ( T , )) → SU (2).To prove the claim for positive r Z , we note that Dehn surgery of slope s = k +1 k on T , gives a lens space, which is SU (2)-abelian, for all integers k ≥ r = mn is another SU (2)-abelian surgery slope for T , , for some fixed r <
3. ThenLin [Lin16] proved that the distance ∆( r, s ) between these slopes is at most the sum of theabsolute values of their numerators, i.e.,(6 k + 1) n − km ≤ (6 k + 1) + m ⇐⇒ ( k + 1) m ≥ (6 k + 1)( n − . Since n ≥
2, the right side is at least (6 k + 1) n , hence r = mn ≥ k + 12( k + 1) = 3 − k + 1) . But this is false for large enough k , so we have a contradiction. (cid:3) We now address Question 1.4, asked by Kronheimer and Mrowka in [KM04a, § SU (2)-abelian, proving Theorem1.8 at the end of this section. We begin with the following observation. Lemma 9.5. If K ⊂ S is a knot for which S ( K ) is SU (2) -abelian then det( K ) = 1 .Proof. Suppose for a contradiction that det( K ) >
1. Klassen [Kla91, Theorem 10] provedthat there are 12 (det( K ) − > ρ : π ( S r K ) → SU (2)with image in the binary dihedral group D ∞ = { e iθ } ∪ { e iθ j } , where here we view SU (2) as the unit quaternions. Letting ρ be any such homomorphism,we observe that ρ ( µ ) cannot lie in the normal subgroup { e iθ } ; otherwise, since the meridian µ normally generates π ( S r K ), this would force the image of ρ to be abelian. So ρ ( µ ) NSTANTONS AND L-SPACE SURGERIES 69 is a purely imaginary quaternion. We therefore have that ρ ( µ ) = − ρ ( µ ) = 1.We also claim that ρ ( λ ) = 1; indeed, the longitude λ belongs to the second commutatorsubgroup of π ( S r K ), so its image lies in D ′′∞ = { e iθ } ′ = { } . Putting all of this together,we have that ρ ( µ λ ) = 1, so ρ induces a representation π ( S ( K )) → SU (2) with the sameimage as ρ , and this is nonabelian, a contradiction. (cid:3) Proposition 9.6.
Suppose K ⊂ S is a nontrivial knot of genus g . Then • S ( K ) is not SU (2) -abelian, and • S ( K ) is not SU (2) -abelian unless K is fibered and strongly quasipositive and g = 2 .Proof. Suppose that S r ( K ) is SU (2)-abelian for r equal to either 3 or 4. Then Corollary 9.2says that S r ( K ) is an instanton L-space, so by Theorem 1.15, K is fibered and stronglyquasipositive and 2 g − ≤ r <
5, i.e., g ≤ g = 1, we note that by Corollary 7.13, K would have to be the right-handedtrefoil. But S ( T , ) is not SU (2)-abelian by Lemma 9.5 since det( T , ) = 3, and neitheris S ( K ) since π ( S ( T , )) is the binary tetrahedral group (see [Rol90, § SU (2). Thus g = 2.In the case r = 3 there is nothing left to prove, so we may now assume that r = 4. Againby Lemma 9.5, we must have det( K ) = 1. Note that K is not the right-handed trefoil since g = 2, which means that S ( K ) and S ( K ) are not instanton L-spaces, by Proposition 7.12.This implies by Proposition 7.11 thatdim I ( S ( K ) , µ ) ≥ . But g = 2 also implies that S ( K ) is also an instanton L-space, by Theorem 1.15, in whichcase another application of Proposition 7.11 tells us that the inequality above is an equality.Since χ ( I ( S ( K ) , µ )) = 0 by Proposition 2.18, we havedim I ( S ( K ) , µ ) = 2 dim I ( S ( K ) , µ ) = 6 . Thus, to demonstrate a contradiction, it will suffice to show that dim I ( S ( K ) , µ ) ≥ K is fibered of genus 2, the Alexander polynomial of K has the form∆ K ( t ) = at + bt + (1 − a − b ) + bt − + at − for some integers a = ± b . We compute from this thatdet( K ) = | ∆ K ( − | = | − b | , and since det( K ) = 1 we must have b = 0. Letting ˆΣ ⊂ S ( K ) be a capped-off Seifert surfacefor K with g ( ˆΣ) = 2, we now apply Theorem 3.6 to see that the Euler characteristics of the(2 j, µ ( ˆΣ) , µ (pt) acting on I ∗ ( S ( K )) µ satisfy X j = − χ ( I ∗ ( S ( K ) , ˆΣ , j ) µ ) t j = ∆ K ( t ) − t − t − = a ( t + 2 + t − ) . The ( ± , , ± i, − , − I ∗ ( S ( K )) µ ≥ . Since g ( ˆΣ) = 2 and µ · ˆΣ = ±
1, we may apply Corollary 2.29 to see thatdim I ( S ( K ) , µ ) = dim I ∗ ( S ( K )) µ ≥ , which is the desired contradiction. (cid:3) Proof of Theorem 1.8.
The slopes r ∈ (2 ,
3) are handled by Proposition 9.4, while Proposi-tion 9.6 addresses r = 3 and r = 4. (cid:3) A -polynomials of torus knots Basic properties of A -polynomials. The goal of this section is to use our precedingresults to prove that a slight enhancement of the A -polynomial detects infinitely many torusknots, including the trefoil, per Theorem 1.11. We begin by recalling the definition of the A -polynomial A K ( M, L ) by Cooper, Culler, Gillet, Long, and Shalen [CCG + X ( K ) denote the variety of characters of representations π ( S r N ( K )) → SL (2 , C ) , and let X ( ∂N ( K )) be the SL (2 , C ) character variety of the boundary torus, with restrictionmap i ∗ : X ( K ) → X ( ∂N ( K )) . The representation variety Hom( π ( T ) , SL (2 , C )) has a subvariety ∆ of diagonal represen-tations, with a branched double covering t : C ∗ × C ∗ ∼ −→ ∆ ։ X ( ∂N ( K ))sending a pair ( M, L ) to the character of the representation ρ with ρ ( µ ) = (cid:18) M M − (cid:19) , ρ ( λ ) = (cid:18) L L − (cid:19) . We let V ⊂ X ( ∂N ( K )) be the union of the closures i ∗ ( X ) as X ranges over irreduciblecomponents of X ( K ) such that i ∗ ( X ) has complex dimension 1. Then V ( K ) = t − ( V ) ⊂ C ∗ × C ∗ is an algebraic plane curve, and we take A K ( M, L ) ∈ C [ M ± , L ± ]to be its defining polynomial. It is normalized to have integer coefficients and no repeatedfactors, and is well-defined up to multiplication by powers of M and L .The A -polynomial of any knot K ⊂ S always has a factor of L −
1. This correspondsto a 1-dimensional curve X red of characters of reducible representations ρ such that ρ ( µ ) = (cid:0) M M − (cid:1) and ρ ( λ ) = ( ) as M ranges over C ∗ ; clearly t − ( i ∗ ( X red )) = C ∗ ×{ } = { L = 1 } .We will work with a slight modification of the A -polynomial, following Ni–Zhang [NZ17]:we define ˜ A K ( M, L ) ∈ Z [ M ± , L ± ]to be the defining polynomial of t − [ X = X red i ∗ ( X ) ⊂ C ∗ × C ∗ , NSTANTONS AND L-SPACE SURGERIES 71 where we take the union over irreducible components X = X red for which dim C i ∗ ( X ) = 1;the difference is that we now explicitly exclude X red , and so there may not be a factor of L −
1. By convention we take ˜ A U ( M, L ) = 1.
Remark 10.1. ˜ A K ( M, L ) is equal to either A K ( M, L ) or A K ( M, L ) / ( L − L − Theorem 10.2. If K is not the unknot, then ˜ A K ( M, L ) has an irreducible factor otherthan L − . Thus A K ( M, L ) = L − iff K is unknotted. Our goal is to prove similar results characterizing torus knots in terms of their A-polynomials. We begin with the following computation.
Proposition 10.3. If K is the ( p, q ) -torus knot, then ˜ A K ( M, L ) divides M pq L − .Proof. The knot complement is Seifert fibered, with generic fiber σ = µ pq λ being central in π ( S r N ( K )). Its image under any representation ρ : π ( S r N ( K )) → SL (2 , C )commutes with the entire image of ρ . If we assume that ρ sends the peripheral subgroup h µ, λ i to diagonal matrices, then ρ ( σ ) is diagonal, and if any other element ρ ( g ) is notdiagonal then this forces ρ ( σ ) = ± I . In other words, we have ρ ( µ pq λ ) = ± I unless ρ isreducible, in which case the character tr( ρ ) lies in X red .If X ⊂ X ( K ) is an irreducible component other than X red satisfying dim i ∗ ( X ) = 1, thenall but at most finitely many χ ∈ X are the characters of irreducible representations ρ : π ( S r N ( K )) → SL (2 , C ) , so that χ ( µ pq λ ) must be identically either 2 or − X , corresponding to ρ ( µ pq λ ) = I or − I respectively. But then t − ( i ∗ ( X )) lies in the zero set of either M pq L − M pq L + 1,from which the proposition follows. (cid:3) Remark 10.4.
By [CCG +
94, Proposition 2.7], the A -polynomial of T p,q has a factor of M pq L + 1, so in light of Proposition 10.3, the only ambiguity in ˜ A T p,q ( M, L ) is whether italso contains a factor of M pq L − K ⊂ S , we will let N ( K ) denote the Newton polygon in the ( L, M )-planeof the polynomial ˜ A K ( M, L ). This is the convex hull of all points ( a, b ) ∈ Z such that themonomial M b L a has nonzero coefficient in ˜ A K ( M, L ); it is well-defined up to translation.We can reinterpret Theorem 10.2 as the statement that N ( K ) is not a single point unless K is the unknot. Definition 10.5.
We say that a nontrivial knot K ⊂ S is r -thin for some r ∈ Q if N ( K )is contained in a line segment of slope r . Example 10.6.
Proposition 10.3 says that N ( T p,q ) is contained in the line segment from(0 ,
0) to (2 , pq ), of slope pq , so T p,q is pq -thin. Proposition 10.7. If K ⊂ S is a hyperbolic knot then K is not r -thin for any r . Proof.
This is essentially [CCG +
94, Proposition 2.6]. The key observation is that X ( K )contains a 1-dimensional irreducible component X , one of whose points is the characterof a discrete faithful representation, such that the function I γ = tr( ρ ( γ )) is not constanton X for any peripheral element γ . If K were pq -thin then ˜ A K ( M, L ) would be (up to amonomial factor) a polynomial of the form f ( M p L q ) for some f ∈ Z [ t ]. But then I M p L q could only take finitely many values on X , corresponding to the roots of f ( t ), and this is acontradiction. (cid:3) SU (2) -averse knots. Sivek and Zentner make the following definition in [SZ17].
Definition 10.8.
A nontrivial knot K ⊂ S is SU (2) -averse if the set S ( K ) = (cid:26) pq ∈ Q (cid:12)(cid:12)(cid:12)(cid:12) S p/q ( K ) is SU (2) − abelian (cid:27) is infinite.One of the main theorems in [SZ17] is the following. Theorem 10.9. If K is SU (2) -averse, then S ( K ) ⊂ R is bounded and has a single accu-mulation point r ( K ) , which is a rational number with | r ( K ) | > . Supposing that r ( K ) ispositive, if we let n = ⌈ r ( K ) ⌉ − then S n ( K ) is an instanton L-space. We call r ( K ) the limit slope of K . The assumption that r ( K ) > K is SU (2)-averse then so is its mirror K , with r ( K ) = − r ( K ).Recall from Theorem 1.10 (proved in the introduction using Theorem 1.15) that if K isan SU (2)-averse knot, then K fibered and strongly quasipositive with r ( K ) > g ( K ) − Proposition 10.10.
Let K ⊂ S be a nontrivial r -thin knot, in the sense of Definition 10.5.Then K is SU (2) -averse with limit slope r . In particular, K is fibered and | r | > g ( K ) − .Proof. Let r = pq with p and q relatively prime. The assumption that K is r -thin says thatup to a monomial factor, we can write˜ A K ( M, L ) = f ( M p L q )for some polynomial f ∈ Z [ t ].If X is an irreducible component of X ( K ), then i ∗ ( X ) has complex dimension either 0 or1, see e.g. [DG04, Lemma 2.1]. Thus if X does not contribute to the plane curve defining˜ A K ( M, L ) then either X is the curve X red of reducible characters or i ∗ ( X ) ⊂ X ( ∂N ( K ))is a point. Since the latter happens for only finitely many X , it follows that the set S ofpoints ( M, L ) ⊂ C ∗ × C ∗ such that • there is an irreducible representation ρ : π ( S r N ( K )) → SL (2 , C ) with ρ ( µ ) = (cid:18) M M − (cid:19) , ρ ( λ ) = (cid:18) L L − (cid:19) ; • ˜ A K ( M, L ) = 0, or equivalently f ( M p L q ) = 0; NSTANTONS AND L-SPACE SURGERIES 73 is finite.Restricting to the subgroup SU (2), every representation ρ : π ( S r N ( K )) → SU (2) isconjugate to one such that ρ ( µ ) = (cid:18) e iα e − iα (cid:19) , ρ ( λ ) = (cid:18) e iβ e − iβ (cid:19) for some constants α and β . (Indeed, every element of SU (2) is diagonalizable, and ρ ( µ )and ρ ( λ ) can be simultaneously diagonalized because they commute.) Let T ⊂ U (1) × U (1) ⊂ C ∗ × C ∗ be the set of all such pairs ( e iα , e iβ ) arising from irreducible SU (2) representations ρ . Thenwe have ˜ A K ( e iα , e iβ ) = f ( e i ( pα + qβ ) ) = 0on all of T except for the finitely many points of S . In particular, e i ( pα + qβ ) can only takefinitely many values on T r S , namely the roots of f ( t ), so [SZ17, Theorem 8.2] tells us that K is SU (2)-averse with limit slope pq = r . We apply Theorem 1.10 to conclude. (cid:3) Finally, we recall the following facts about SU (2)-averse satellite knots, proved in [SZ17,Theorem 1.7]. Theorem 10.11.
Let K = P ( C ) be a nontrivial, SU (2) -averse satellite, and let w ≥ bethe winding number of the pattern P ⊂ S × D . • If P ( U ) is not the unknot, then it is SU (2) -averse with limit slope r ( K ) . • If w = 0 , then the companion C is SU (2) -averse, and r ( K ) = w r ( C ) . Detecting torus knots.
Ni and Zhang [NZ17] proved that the combination of thepolynomial ˜ A K ( M, L ) and the knot Floer homology [ HFK ( K ) suffice to detect torus knots.The reason they needed [ HFK ( K ) was to show that K is fibered, and to determine its Seifertgenus and Alexander polynomial. However, we have seen that the ( p, q )-torus knot T p,q is pq -thin, and that r -thin knots are fibered for any r , so in many cases we do not actuallyneed [ HFK ( K ). We make this precise below. Lemma 10.12.
Suppose that K is an r -thin knot for some r ∈ Q , but that K is not isotopicto a torus knot. Then K is SU (2) -averse with limit slope r , and it is both fibered and asatellite knot. If we write K = P ( C ) then • the satellite pattern P ⊂ S × D has positive winding number w ≥ ; • the companion C is fibered and rw -thin; and • the knot P ( U ) is fibered, and if it is not the unknot then it is r -thin; • the Seifert genera of K , P ( U ) and C are related by g ( K ) = w · g ( C ) + g ( P ( U )) . Moreover, if P ( U ) is the unknot then w ≥ .Proof. Proposition 10.10 says that K is fibered. In addition, K is not the unknot by The-orem 10.2 or a torus knot by assumption, and hyperbolic knots are not r -thin by Proposi-tion 10.7, so K must be a satellite.For the claims that w ≥ C and P ( U ) are both fibered, we note by [BZ03,Proposition 5.5] that since K is fibered, the commutator subgroup of π ( S r N ( K )) is free on 2 g ( K ) generators. Since it is finitely generated, we conclude from [BZ03, Corollary 4.15]that the winding number must be nonzero. It follows from Theorem 10.11 that C is SU (2)-averse, and that either P ( U ) is the unknot or it is also SU (2)-averse, and in any caseProposition 10.10 tells us that these must both be fibered. Now we have the relation∆ K ( t ) = ∆ P ( U ) ( t ) · ∆ C ( t w ) , and since each of these knots is fibered the claim about their Seifert genera follows bycomputing the degrees of each of these Alexander polynomials.For the thinness of C and P ( U ), Ni–Zhang [NZ17, Lemma 2.6] proved that ˜ A P ( U ) ( M, L )divides ˜ A K ( M, L ) in Z [ x, y ], so if P ( U ) is not the unknot (meaning that ˜ A P ( U ) ( M, L ) = 1by Theorem 10.2) then it is also r -thin. They also proved [NZ17, Proposition 2.7] that everyirreducible factor f C ( M, L ) | ˜ A C ( M, L )contributes a factor f K ( M, L ) = ( Red (cid:2)
Res ¯ L (cid:0) f C ( M w , ¯ L ) , ¯ L w − L (cid:1)(cid:3) , deg L f C ( M, L ) > f C ( M w , L ) , deg L f C ( M, L ) = 0to ˜ A K ( M, L ). Here Res ¯ L denotes the resultant that eliminates the variable ¯ L , and thereduced polynomial Red( p ( M, L )) is obtained from p ( M, L ) by removing all repeated factors.Since K is r -thin, the case deg L f C ( M, L ) = 0 does not occur and it follows that C is rw -thin.Finally, suppose that P ( U ) = U but that w = 1. Since K is fibered, Hirasawa, Murasugi,and Silver [HMS08, Corollary 1] proved in this case that the pattern P must be the coreof S × D , contradicting the fact that K is a nontrivial satellite, so we must have w ≥ P ( U ) = U and w = 2 then they proved that P must be a ( ± , K = P ( C ) cannot be SU (2)-averse, so this is also impossible. We conclude that if P ( U ) = U then w ≥ (cid:3) Proposition 10.13.
Suppose that K is an r -thin knot for some r ∈ Q . Then r is a nonzerointeger with at least two distinct prime divisors; in other words, r = pq for some nontrivialtorus knot T p,q .Proof. We may assume that r is nonnegative, by replacing K with its mirror K if needed.Proposition 10.10 says that K is SU (2)-averse with limit slope r , hence by Theorem 10.9we know that S n ( K ) is an instanton L-space where n = ⌈ r ⌉ −
1. If 0 ≤ r ≤ S ( K ) is an instanton L-space, so K is the right-handed trefoil by Proposition 7.12;but then we should have r = 6, so this is a contradiction and in fact r > R = { r ∈ Q | r > , r − thin knots exist , r = pq for any T p,q } is nonempty, and let r = inf R ; then r ≥ r ∈ R such that r ≤ r < r + 1, and we choose K to have minimal Seifert genus among all r -thinknots; by definition K is not a torus knot.By Lemma 10.12 we can write K as a satellite P ( C ), and then g ( K ) = w · g ( C ) + g ( P ( U )) . NSTANTONS AND L-SPACE SURGERIES 75
Thus g ( P ( U )) < g ( K ), and if P ( U ) = U then we know that P ( U ) is also r -thin, contra-dicting the minimality of g ( K ). We must therefore have P ( U ) = U , and so Lemma 10.12says that w ≥ K is r -thin and w ≥
3, the companion knot C has thinness rw ≤ r < r + 19 < r , the last inequality holding since r ≥ > . If r is not an integer with at least two distinctprime factors then neither is rw , so rw ∈ R ; but since rw < inf R we have a contradiction.So in fact K cannot exist, and we conclude that the set R is empty, as desired. (cid:3) We can now prove that ˜ A K ( M, L ) detects the trefoils.
Theorem 10.14. If K ⊂ S is -thin, then K is isotopic to the right-handed trefoil.Proof. Suppose that K is 6-thin but not isotopic to T , , and that K minimizes Seifertgenus among such knots. In this case Lemma 10.12 says that K is a satellite of the form K = P ( C ) with winding number w ≥
1, and that C is w -thin; by Proposition 10.13 thisforces w = 1. Given this, Lemma 10.12 now also implies that P ( U ) is a nontrivial, 6-thinknot, and that g ( K ) = g ( C ) + g ( P ( U )) . Since g ( K ) was assumed minimal, both C and P ( U ) must be isotopic to T , . Thus K isan instanton L-space knot of genus 2, with∆ K ( t ) = ∆ P ( U ) ( t )∆ C ( t w ) = ( t − t − ) , or equivalently(10.1) ∆ K ( t ) − t − t − = t + t − . Let ˆΣ ⊂ S ( K ) be a capped-off Seifert surface for K with g ( ˆΣ) = 2. Then Theorem 3.6together with (10.1) tells us that the (2 j, µ ( ˆΣ) , µ (pt) acting on I ∗ ( S ( K )) µ have Euler characteristics given by χ ( I ∗ ( S ( K ) , ˆΣ , j ) µ ) = ( j = ± j = ± . The ( ± , φ ), there-fore have the same dimension 1 + 2 m for some integer m ≥
0, and the (0 , k for some integer k ≥
0. Since these eigenspaces are isomorphic to the( ± i, − , − I ∗ ( S ( K )) µ = 4 + 8 m + 4 k. Since g ( ˆΣ) = 2 and µ · ˆΣ = ±
1, we may apply Corollary 2.29 to conclude thatdim I ( S ( K ) , µ ) = dim I ∗ ( S ( K )) µ = 4 + 8 m + 4 k. Together with the fact that χ ( I ( S ( K ) , µ )) = 0, this impliesdim I ( S ( K ) , µ ) = 12 dim I ∗ ( S ( K )) µ = 2 + 4 m + 2 k. Now, Proposition 7.11 says that S g ( K ) − ( K ) = S ( K ) is an instanton L-space, which thenimplies by the same proposition that3 ≥ dim I ( S ( K ) , µ ) = 2 + 4 m + 2 k, so m = k = 0 and dim I ( S ( K ) , µ ) = 2 . But the latter implies by Proposition 7.11 that S ( K ) is an instanton L-space, and Propo-sition 7.12 tells us in this case that K is the unknot or the right-handed trefoil, a contra-diction. (cid:3) Proposition 10.15.
Let r > be an integer such that one of the following holds: • r is square-free and odd, with at least two distinct prime divisors; • r = p q for distinct primes p and q , with q ≥ ; or • r = p q for distinct primes p and q .Then every r -thin knot is a torus knot.Proof. Suppose that there are non-torus knots which are r -thin, and let K be such a knotwith the smallest possible genus. By Lemma 10.12, we know that K is a satellite, say K = P ( C ) where P has winding number w ≥
1, and that the companion C is rw -thin. If w ≥ rw -thin knots, so we must have w = 1.Since w = 1, we know from Lemma 10.12 that P ( U ) is not the unknot, that both C and P ( U ) are r -thin, and that g ( K ) = g ( C ) + g ( P ( U )) . Both g ( C ) and g ( P ( U )) are positive and strictly less than g ( K ), but g ( K ) was assumedminimal, so C and P ( U ) must be nontrivial torus knots, say C = T a,b and P ( U ) = T c,d with ab = cd = r .The assumptions on r each imply that a, b ≥
3, so we have g ( C ) = ( a − b − ab + 12 − a + b ≥ r + 12 − r/
32 = r − . Likewise g ( P ( U )) satisfies the same bound, so g ( K ) = g ( C ) + g ( P ( U )) satisfies2 g ( K ) − ≥ (cid:18) r − (cid:19) − r − . Since K is r -thin we know from Proposition 10.10 that r > g ( K ) −
1, or equivalently r − ≥ g ( K ) − r is an integer. But then we have r − ≥ r −
5, or r ≤
12, andthis is a contradiction. (cid:3)
We say that the ˜ A -polynomial detects a knot K if for any knot K ′ , we have ˜ A K ( M, L ) =˜ A K ′ ( M, L ) iff K is isotopic to K ′ . Corollary 10.16.
Let p and q be distinct odd primes. Then the ˜ A -polynomial detects eachof the torus knots T p,q , T p ,q , T p ,q , T ,q if q > , and T ,q .Proof. Write any of the given torus knots as T a,b . Then r = ab satisfies the hypotheses ofProposition 10.15, so if ˜ A K ( M, L ) = ˜ A T a,b ( M, L ) then K must be a torus knot T c,d with r = cd . But in each case T a,b is the unique r -thin torus knot, so in fact K = T a,b . (cid:3) NSTANTONS AND L-SPACE SURGERIES 77
Proof of Theorem 1.11.
The trefoil case is Theorem 10.14, and the remaining cases areincluded in Corollary 10.16. (cid:3)
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Department of Mathematics, Boston College
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