Deep and shallow slice knots in 4-manifolds
DDEEP AND SHALLOW SLICE KNOTS IN 4-MANIFOLDS
MICHAEL R. KLUG AND BENJAMIN M. RUPPIK
Abstract.
We consider slice disks for knots in the boundary of a smooth compact 4-manifold X . We call a knot K ⊂ ∂X deep slice in X if there is a smooth properly embedded 2-disk in X with boundary K , but K is not concordant to the unknot in a collar neighborhood ∂X × I of the boundary.We point out how this concept relates to various well-known conjectures and give some criteriafor the nonexistence of such deep slice knots. Then we show, using the Wall self-intersectioninvariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzeronumber of 2-handles always has a deep slice knot in the boundary.We end by considering 4-manifolds where every knot in the boundary bounds an embeddeddisk in the interior. A generalization of the Murasugi-Tristram inequality is used to show thatthere does not exist a compact, oriented 4-manifold V with spherical boundary such that everyknot K ⊂ S = ∂V is slice in V via a null-homologous disk. Overview
The Smooth 4-Dimensional Poincaré Conjecture (SPC4) proposes that every closed smooth4-manifold Σ that is homotopy equivalent to S is diffeomorphic to the standard S . By workof Freedman [Fre82], it is known that if Σ is homotopy equivalent to S , then Σ is in facthomeomorphic to S . In stark contrast to the SPC4, it might be the case that every compactsmooth 4-manifold admits infinitely many distinct smooth structures. The existence of an exotichomotopy 4-sphere is equivalent to the existence of an exotic contractible compact manifoldwith S boundary [Mil65, p. 113], henceforth called an exotic homotopy 4-ball .One possible approach to proving that a proposed exotic homotopy 4-ball B is in fact exoticis to find a knot K ⊂ S = ∂ B , such that there is a smooth properly embedded disk D , → B ,with ∂D mapped to K , where K is not smoothly slice in the usual sense in the standard4-ball B . A knot is (topologically/smoothly) slice in B if and only if it is null-concordant in S × I = S × [0 , S × I , → S × I whose oriented boundary is K ⊂ S × { } together with the unknot U ⊂ S × { } . Anotherway of thinking about this strategy is that we want to find a knot K in S = ∂ B that bounds aproperly embedded smooth disk in B but does not bound any such disk that is contained ina collar S × I of the boundary of B . In this case, to verify the sliceness of K , we have to go“deep” into B .An easier task might be to find a homology 4-ball X with S boundary such that there isa knot in the boundary that bounds a smooth properly embedded disk in X but not in B ,however, this is also an open problem. In [FGMW10], the authors investigate the possibility ofproving that a homotopy 4-ball B with S boundary is exotic by taking a knot in the boundarythat bounds a smooth properly embedded disk in B and computing the s -invariant of K , inthe hopes that s ( K ) = 0, whereby they could then conclude that B is exotic. Unfortunatelyfor this approach as noted in the paper, it turns out that the homotopy 4-ball that they werestudying was in fact diffeomorphic to B , see [Akb10]. It is still open whether the s -invariant Mathematics Subject Classification.
Key words and phrases.
Concordance in general 4-manifolds, Slice disks, Murasugi-Tristram-Inequality.MK and BR were supported by the Max Planck Institute for Mathematics in Bonn. a r X i v : . [ m a t h . G T ] O c t MICHAEL R. KLUG AND BENJAMIN M. RUPPIK
K∂X ? collar X ∆ Figure 1.
Schematic of a deep slice disk ∆ (blue) in a 4-manifold X , withboundary the knot K ⊂ ∂X . The knot K is called deep slice if it does not bounda properly embedded disk in a collar of the boundary, indicated by the (lightblue) dashed lines.can obstruct the sliceness of knots in B that are slice in some homotopy 4-ball, as is noted inthe corrigendum to [KM13].Motivated by this, we make the following definitions: For a 3-manifold M containing aknot K : S , → M , we say that K is null-concordant in M × I if there is a smoothly properlyembedded annulus S × I , → M × I cobounding K ⊂ M × { } on one end and an unknotcontained in a 3-ball U ⊂ B ⊂ M × { } on the other. Equivalently, K ⊂ M × { } bounds asmoothly properly embedded disk in M × I . Definition 1.1 (Deep slice/Shallow slice) . Let X be a smooth compact 4-manifold withnonempty boundary ∂X . We call a knot K ⊂ ∂X deep slice in X if there is a smooth properlyembedded disk in X with boundary K , but K is not null-concordant in a collar neighborhood ∂X × I of the boundary.If K is slice in X but not deep slice, we will call it shallow slice in X – this is equivalent to K being null-concordant in the collar ∂X × I . See Figure 1 for a schematic illustration of thesedefinitions.In this language, Problem 1.95 on Kirby’s list [Kir95] (attributed to Akbulut) can be reformu-lated as follows: Are there contractible smooth 4-manifolds with boundary an integral homology3-sphere which contain deep slice knots that are null-homotopic in the boundary? Note that anyknot that is not nullhomotopic in the boundary will not be shallow slice and thus if it is slice, itwill be deep slice. For this reason we will be looking for deep slice knots that are null-homotopicin the boundary. We will often consider our knots to be contained in 3-balls in the boundary,which we call local knots , so we can freely consider them in the boundary of any 4-manifold anddiscuss if they are slice there. To avoid confusion when we say that a (local) knot in a 3-manifold M is slice we will usually qualify it with “in X ”.1.1. Outline.
In the first part of this paper we will restrict ourselves to the smooth category,starting in Section 2, where we discuss a condition that guarantees that some 4-manifolds haveno deep slice knots and related results. In Section 3, we prove that every 2-handlebody has a
EEP AND SHALLOW SLICE KNOTS IN 4-MANIFOLDS 3 deep slice knot in its boundary. To do this we employ the Wall self-intersection number and aresult of Rohlin which we discuss briefly.In Section 4, we recall the Norman-Suzuki trick and observe that every 3-manifold bounds a4-manifold where every knot in the boundary bounds a properly embedded disk. In contrast,if we restrict to slice disks trivial in relative second homology, we will see that every compacttopological 4-manifold with boundary S contains a knot which does not bound a null-homologoustopological slice disk. We finish with some questions and suggestions for further directions inSection 5.1.2. Conventions.
In the literature, properly embedded slice disks in a 4-manifold X are oftenassumed to be null-homologous in H ( X, ∂X ). We will make this extra assumption on homologyonly in Section 4 when discussing the “universal slicings”. For the first part deep slice and shallow slice will describe the existence of a embedded disks with the relevant properties withoutconditions on the homology class.Starting from an n -manifold M n without boundary, we obtain a punctured M (more preciselya bounded punctured M ) by removing a small open n -ball M ◦ := M \ int D n , which yields amanifold with boundary ∂M ◦ = S n − . Observe that a punctured M is the same as a connectedsum M ◦ ∼ = M D n with a n -ball.1.3. Acknowledgments.
The authors would like to thank Anthony Conway, Rob Kirby, MarkPowell, Arunima Ray and Peter Teichner for helpful conversations, their encouragement andguidance. BR would like to thank Thorben Kastenholz for asking about the decidability of theembedded genus problem in 4-manifolds, which motivated Section 4. We are especially gratefulto Akira Yasuhara for pointing us to related literature and the work of Suzuki. The Max PlanckInstitute for Mathematics in Bonn supported us with a welcoming research environment.2.
Nonexistence of deep slice knots
For starters, we have:
Proposition 2.1.
There are no deep slice knots in \ k S × B .Proof. Let K ⊂ k S × S = ∂ ( \ k S × B ) such that K is slice in \ k S × B . Then, thinkingof \ k S × B as a wedge of k copies of S thickened to be 4-dimensional, if D is any slice diskfor K we can isotope D such that it does not intersect a one-dimensional wedge of circles that \ k S × B deformation retracts onto. Therefore, D can be isotoped to be contained in a collarneighborhood of the boundary k S × S and thus K is shallow slice. (cid:3) The following might be a surprise, as one could expect that additional topology in a 3-manifold M creates more room for concordances: Proposition 2.2 (Special case of [NOPP19, Prop. 2.9]) . If a local knot K ⊂ B ⊂ M is nullconcordant in M × I , then K is null concordant in S × I .Proof sketch. Let D be a properly embedded disk in M × I with boundary K and let f M be theuniversal cover of M . Then D lifts to a properly embedded disk e D ⊂ f M × I . Further, since K is contained in a 3-ball B , all of the lifts of K to f M are just copies of K , and therefore, theboundary of e D is a copy of K , considered inside of f M . As a consequence of geometrization[Per03], we know that every universal cover of a punctured compact 3-manifold smoothly embedsinto S , as was observed in [BN17, Lem. 2.11]. It follows then that there is an embedding f M × I , → S × I . But then the image of e D under this embedding shows that K bounds a diskin S × I . (cid:3) We have added a proof of this proposition here to highlight that this lifting argumentbreaks down in the case of higher genus surfaces if their inclusion induces a nontrivial mapon fundamental groups. If K bounds a genus g surface with one boundary component Σ g, in MICHAEL R. KLUG AND BENJAMIN M. RUPPIK M × I , we can only lift this to the universal cover (and subsequently find a genus g surface for K in S via this method) under the condition that the inclusion of Σ g, in M × I is π -trivial.So this argument does not work if the surface really “uses the extra topology of M ”. Example 2.3.
Take a non-orientable 3-manifold M containing the connected sum K K oftwo copies of a local invertible knot K with smooth 4-ball genus g ( K K ) ≥
2. As an explicitexample, K a left-handed trefoil will work, and we illustrate the following in Figure 2. Describean embedded torus in M × I with the motion picture method: Use the connected sum bandto split the sum with a saddle. Then let one of the summands travel around an orientationreversing loop in M while leaving the other one fixed. The summand traveling around theloop was reflected in the process and since it is invertible it is isotopic to − K = rK in a 3-ballneighborhood in M . Fusing the summands back together along a connected sum band we nowobtain K − K as a local knot. Finally cap this off with the usual ribbon disk for the connectedsum of a knot with its concordance inverse. The 4-ball genus g ( K K ) ≥ M × I ) -4-genus , which we define as g M × I ( J ) := min { g | ∃ smooth proper embedding Σ g, , → M × I with ∂ Σ g, = J ⊂ M × { }} Observe that in this notation the usual 4-ball genus is g = g S × I and we can rephraseProposition 2.2 as g M × I ( K ) = 0 implies g ( K ) = 0 for local knots K . Similar notions of 4-generawere introduced in Celoria’s investigation of almost-concordance [Cel18, Def. 12]. (a) Saddle move to separate the summands of K K . (b) One of the summands travels around an orienta-tion reversing loop in M . (c) It returns mirrored, now add a fusion band. (d)
Finish off the movie with the standard ribbondisk for K − K . Figure 2.
Four frames of the movie of a properly embedded punctured torus in M × I with boundary K K ⊂ M ×{ } , where M is a non-orientable 3-manifold.It would be interesting to find an example of an orientable 3-manifold M where the g M × I ( K )genus of some local knot K ⊂ D ⊂ M is strictly smaller than the 4-ball genus g ( K ), or provethat no such M exists. Local K satisfy g M × I ( K ) ≤ g ( K ) as cobordisms in S × I can beembedded into M × I . Because of Proposition 2.2 an example where these values differ can onlyappear for g ( K ) ≥
2. Moreover, as we will see in Proposition 2.4 such an M would necessarily EEP AND SHALLOW SLICE KNOTS IN 4-MANIFOLDS 5 not embed in S . Another special case is treated in [DNPR18, Thm. 2.5] where a handlecancellation argument shows that there is no difference for local knots in M = S × S , that isthe equality g S × S × I = g holds (and also analogous statements for k S × S ). Topologicalconcordance in S × S × I is investigated in [FNOP19].We now give a criterion that shows that certain 4-manifolds have no local deep slice knots inthe boundary. This idea is also contained in [Suz69, Thm. 0] and its variants. Proposition 2.4.
Let X be a compact 4-manifold with a local knot γ ⊂ B ⊂ ∂X that is slicein X . If there is a cover of X which can be smoothly embedded into S , then γ ⊂ B , → S = ∂B is slice in B . Hence, γ is shallow slice in X .Proof. Let e X be a cover of X with an embedding e X ⊂ S into S and let e D be a lift of a slicedisk for γ to e X with e γ = ∂ e D . Note that the knot e γ is the same as γ , since γ is contained in a3-ball and the only covers of a 3-ball are disjoint unions of 3-balls. Puncture S by removinga small ball B close to e γ and such that e γ can be connected by an annulus disjoint from ˜ X to ∂B and such that the other end of the annulus is (the mirror image of) K ⊂ ∂B . Then since S − int B ∼ = B , the annulus together with e D show that K is slice in the B which is thecomplement of the small ball. Therefore γ is shallow slice in X . (cid:3) As an example, Proposition 2.4 implies that \ k S × D contains no deep slice local knots,since these manifolds can all be embedded in S . However, these manifolds all contain deep sliceknots, necessarily non-local, as will be seen shortly. Additionally, we have: Corollary 2.5.
Suppose that X is a closed smooth 4-manifold with universal cover R or S ,and let X ◦ denote the punctured version. Then X ◦ has no deep slice knots. Existence of deep slice knots A is a 4-manifold whose handle decomposition contains one 0-handle, somenonzero number of 2-handles and no handles of any other index. In this section we prove: Theorem 3.1.
Every 2-handlebody X contains a null-homotopic deep slice knot in its boundary. Remark 3.2.
For the special case of the 2-handlebody D × S the existence of such knots wasalready observed in [DNPR18, Thm. B] (here only winding number w = 0 gives null homotopicknots). Furthermore the authors construct an infinite family of slice knots which are pairwisedifferent in topological concordance in a collar of the boundary.Theorem 3.1 breaks up naturally into two cases depending on whether the boundary hasnontrivial π or not (i.e. if it is or is not S ). In the case where π ( ∂X ) = 1, there is a concordanceinvariant for knots in arbitrary 3-manifolds, closely related to the Wall self-intersection number(see [Wal99], [FQ90], and [Sch03]), that will allow us to show that some obviously slice knotsare not shallow slice. In the case where π ( ∂X ) is trivial, and therefore by the 3-dimensionalPoincaré conjecture [Per03] ∂X = S , the Wall self-intersection number is of no use. However,in this case, the consideration of whether a knot that is slice in X is deep slice in X is relatedto the existence of spheres representing various homology classes in the manifold obtained byclosing X off with a 4-handle. Remark 3.3.
If there was a direct proof that every closed homotopy 3-sphere smoothly bounds acontractible 4-manifold, then we would not need to invoke the 3-dimensional Poincaré conjecture.Following [Yil18] and [Sch03], we briefly introduce the Wall self-intersection number in thesetting that we will be working in, and state some of its basic properties. Let Y be a closedoriented 3-manifold and let γ : S , → Y be a knot in Y . Let C γ ( Y ) denote the set of concordanceclasses of oriented knots in Y that are freely-homotopic to γ . In particular C U ( Y ) denotes theset of concordance classes of oriented null-homotopic knots in Y , where we write U for the localunknot in Y . Given an oriented null-homotopic knot K ⊂ Y , by transversality there exists an MICHAEL R. KLUG AND BENJAMIN M. RUPPIK L i γ ? Wh( γ ) = KL i ? Figure 3.
The Whitehead double of a nontrivial meridian γ to one of the surgerylink components is deeply slice in X .oriented immersed disk D in Y × I with boundary K ⊂ Y × { } = Y that has only double pointsof self-intersection. Let ? ∈ Y denote a basepoint which we implicitly use for π ( Y ) = π ( Y × I )throughout. Choose an arc, which we will call a whisker, from ? to D . For each double point ofself-intersection p ∈ D choose a numbering of the two sheets of D that intersect at p . Then let g p ∈ π ( Y ) be the homotopy class of the loop in Y × I obtained by starting at ? , taking thewhisker to D , taking a path to p going in on the first sheet, taking a path back to where thewhisker meets D that leaves p on the second sheet, and then returning to ? using the whisker.Note that changing the order of the two sheets would transform g p to g − p . Also, since K and Y are oriented, D and Y × I obtain orientations with the convention that K ⊂ Y × { } = Y ,and therefore, for every self-intersection point p ∈ D , there is an associated sign which we willdenote by sign( p ).Let e Λ := Z [ π ( Y )] h{ g − g − | g ∈ π ( Y ) }i ⊕ Z [1]were the quotient is a quotient as abelian groups. The Wall self-intersection number of K isdefined to be µ ( K ) = X p sign( p ) · g p ∈ e ΛSee [Sch03] for a proof that it is independent of the choice of D , the choice of whisker, and thechoice of orderings of the sheets of D around the double points. Further, µ is a concordanceinvariant in Y × I , and therefore defines a map: µ : C U ( Y ) → e ΛNotice that if g ∈ π ( Y ) and g = 1 then g is also nonzero in e Λ. Proof of Theorem 3.1, Case 1.
We are now in position to handle Theorem 3.1 in the case where π ( ∂X ) = 1. Now X is described by attaching 2-handles to D along some framed link L ⊂ ∂D .Since π ( ∂X ) is (normally) generated by the meridians of L and π ( ∂X ) = 1, there is somemeridian γ of L that is nontrivial in π ( ∂X ). Notice that if we are given a 2-handlebodydescribed by a framed link L and K is a knot in the boundary of the 2-handlebody that isshown in the framed link diagram as an unknot (possibly linked with L ), then K is slice in the2-handlebody – just forget all of the other 2-handles and take an unknotting disk whose interiorhas been pushed into the 0-handle. Now, take K in ∂X to be a Whitehead double of γ as inFigure 3, which is a null-homotopic knot in the boundary. By the previous observation, since K is unknotted in the boundary of the 0-handle, K is slice in X . Additionally, one computesthat µ ( K ) = γ = 1 ∈ e Λ, for example using the null-homotopy in Figure 4. Therefore, K is notnull-concordant in ∂X , so K is deep slice in X . (cid:3) Notice that if π ( ∂X ) = 1, then µ is of no use since e Λ = 0. Now assume that π ( ∂X ) = 1so that ∂X = S . Again X is obtained by attaching 2-handles to some framed link L . Let b X EEP AND SHALLOW SLICE KNOTS IN 4-MANIFOLDS 7
Figure 4.
Track of a homotopy from the Whitehead double of γ to the unknotgiving an immersed disk with a single double point (red in the middle frame).The red double point loop based at the green basepoint calculates that µ ( K ) = γ .denote the closed 4-manifold obtained by closing off X with a 4-handle. We will need a lemmaon surfaces in 2-handlebodies, whose statement is standard and could alternatively be concludedfrom the KSS-normal form for surfaces as in [Kam17, Thm. 3.2.7] and [KSS82]. Lemma 3.4.
Let X be a closed smooth 4-manifold with a handle decomposition consistingof only 0-, 2-, and 4-handles, with exactly one 0-handle and one 4-handle. Every element of H ( X ; Z ) can be represented by a smooth closed orientable surface whose intersection with theunion of the 0- and 2-handles of X is a single disk.Proof. Let X ≤ denote the union of the 0- and 2-handles of X , so that X = X ≤ ∪ B . Forevery 2-handle h i , there is an element H ( X ; Z ) obtained by taking the co-core disk D i for h i and capping it off with an orientable surface in the 4-handle. Let { F i } denote a choice of thesesurfaces, one for each 2-handle. These surfaces form a basis for H ( X ; Z ) and note that each hasthe desired property that F i ∩ X ≤ = D i is a disk. Given an arbitrary element x ∈ H ( X ; Z ),we have x = a [ F ] + · · · + a n [ F n ] for some a i ∈ Z . Therefore, by taking parallel copies of the F i for each summand, we can find an immersed (possibly disconnected) orientable surface F representing x , with F ∩ X ≤ a union of P | a i | disjoint disks. By taking arcs in ∂X ≤ thatconnect the different boundaries of the disks all together, and attaching tubes to F along thesearcs, we obtain a connected orientable immersed surface F representing x whose intersectionwith X ≤ is now a disk. To make F into an embedded surface, we can resolve the double pointsin the 4-handle, by increasing the genus, and arrive at a surface representing x with the desiredproperty. (cid:3) The main ingredient for the proof of the second case of Theorem 3.1 is the following theoremof Rohlin, and in particular the corollary that follows. Rohlin’s theorem has been used in asimilar way to study slice knots in punctured connected sums of projective spaces, for examplein [Yas91] and [Yas92].
Theorem 3.5 (Rohlin, [Roh71]) . Let X be an oriented closed smooth 4-manifold with H ( X ; Z ) =0 . Let ψ ∈ H ( X ; Z ) be an element that is divisible by 2, and let F be a closed oriented surfaceof genus g smoothly embedded in X that represents ψ . Then g ≥ | ψ · ψ − σ ( X ) | − b ( X ) Corollary 3.6.
Let X be a closed smooth 4-manifold with H ( X ; Z ) = 0 , and H ( X ; Z ) = 0 .Then there exists a homology class ψ ∈ H ( X ; Z ) that cannot be represented by a smoothlyembedded sphere.Proof of Corollary 3.6. To apply Theorem 3.5, we must find a homology class ψ that is divisibleby 2 where the right hand side | ψ · ψ − σ ( X ) | − b ( X ) >
0. Since the intersection form on X is unimodular, there exists some element α with α · α = 0. Then by taking k to be a sufficientlylarge integer, we can make | (2 kα ) · (2 kα ) − σ ( X ) | arbitrarily large. By taking ψ = 2 kα , theresult follows. (cid:3) MICHAEL R. KLUG AND BENJAMIN M. RUPPIK ∂DS × I X = b X ≤ D Figure 5.
Schematic of the blue surface F in the 2-handlebody, intersecting X = the union of the 0- and 2-handles in a disk D . If ∂D was shallow slice(dashed light blue) in X , disk D union the shallow slice disk flipped into the4-handle (solid light blue) would be an impossible sphere representative of thehomology class of F . Proof of Theorem 3.1, Case 2.
By Corollary 3.6, let ψ ∈ H ( b X ; Z ) be a homology class thatcan not be represented by an embedded sphere. Using Lemma 3.4, let F be a smooth closedorientable surface representing ψ whose intersection with X = b X ≤ is a disk D , as illustratedschematically in Figure 5. Then ∂D ⊂ ∂X is deep slice in X , since otherwise the surface obtainedby intersecting F with the 4-handle could be replaced with a disk without altering the homologyclass, violating the assumption that ψ cannot be represented by an embedded sphere. To seethat the homology class is not altered, observe that in any 2-handlebody the homology class ofa surface is determined by how it intersects the 0- and 2-handles. Also observe that in this casethe deep slice knot ∂D ⊂ ∂X = S is local. This concludes the proof of Theorem 3.1. (cid:3) Universal slicing manifolds do not exist
The Norman-Suzuki trick [Nor69, Cor. 3], [Suz69, Thm. 1] can be used to show that any knot K ⊂ S bounds a properly embedded disk in a punctured S × S : The track of a null-homotopyof K in D can be placed in the punctured S × S which gives a disk that we can assumeto be a generic immersion, missing S ∨ S ⊂ ( S × S ) ◦ , and with a finite number of doublepoints. By tubing into the spheres S × { pt } , { pt } × S we can remove all the intersections –but observe that this changes the homology class of the disk. Proposition 4.1.
Let M be a closed orientable 3-manifold. There exists a compact orientable4-manifold X constructed with only a 0-handle and 2-handles, with ∂X = M such that everyknot in M is slice in X .Proof. Start by taking any compact 4-manifold X with only 0-, 2-handles and boundary M andlet X = X S × S ). Let K ⊂ M = ∂X be a knot. Since X and X are simply connected, K bounds an immersed disk which we can assume lives completely in the X -summand of EEP AND SHALLOW SLICE KNOTS IN 4-MANIFOLDS 9 the connected sum. Now the Norman-Suzuki trick works to remove intersection points of theimmersion by tubing into the coordinate spheres of the S × S -summand. (cid:3) Remark 4.2.
In contrast to the homologically nontrivial disks constructed in the Norman-Suzuki trick, a knot is slice via a null-homologous disk in some connected sum n S × S if andonly if its Arf-invariant is zero. Arf K = 0 implies that the knot is band-pass equivalent to theunknot, and a band pass can be realized by sliding the (oppositely oriented) strands of a pairof bands over the coordinate spheres in a S × S factor. Conway-Nagel [CN20] defined andstudied the minimal number of summands needed to find a disk in a punctured n S × S . Convention:
From now until the end of this section, properly embedded slice disks ∆ ⊂ X in a 4-manifold are always required to be null-homologous . We will still add the qualifier“null-homologous” in the statements to emphasize this. Since our obstructions work in thetopologically locally flat category, we will formulate everything in this more general setting. Definition 4.3.
A knot K ⊂ S is (topologically/smoothly) null-homologous slice in the (topolog-ical/smooth) -manifold X with ∂X = S , if K = ∂ ∆, where ∆ ⊂ X is a (locally flat/smooth)properly embedded disk such that [∆ , ∂ ∆] = 0 ∈ H ( X, ∂X ).One way of studying if a knot K is slice in D is to approximate D by varying the 4-manifold X . By restricting the intersection form and looking at simply-connected 4-manifolds X thisgives rise to various filtrations of the knot concordance group (notably the ( n )-solvable filtration F n of Cochran-Orr-Teichner [COT03] and the positive and negative variants P n , N n [CHH13]).We say that the properly embedded disk ∆ is null-homologous if its fundamental class [∆ , ∂ ∆] ∈ H ( X, ∂X ) is zero. Since by Poincaré duality the intersection pairing H ( X ) ⊗ Z H ( X, ∂X ) (cid:116) −→ Z is non-degenerate, a null-homologous disk is characterized by the property that it intersects allclosed second homology classes algebraically zero times. For slicing in arbitrary 4-manifolds, wehere restrict to null-homologous disks to exclude constructions as in the Norman-Suzuki trick.For every fixed knot K ⊂ S , there is a 4-manifold in which K is null-homologically slice.Norman [Nor69, Thm. 4] already observes that it is possible to take as the 4-manifold a puncturedconnected sum of the twisted 2-sphere bundles S e × S . Similarly, [CL86, Lem. 3.4] discuss thatfor any knot K ⊂ S there are numbers p, q ∈ N such that K is null-homologous slice in thepunctured connected sum p CP q CP of complex projective planes. The argument starts witha sequence of positive and negative crossing changes leading from K to the unknot, and thenrealizes say a positive crossing change by sliding a pair of oppositely oriented strands over the CP in a projective plane summand. The track of this isotopy, together with a disk boundingthe final unknot gives a motion picture of a null-homologous slice disk. Since both positive andnegative crossing changes might be necessary, it is important that both orientations CP , CP are allowed to appear in the connected sum.In view of ( S × S ) ◦ where every knot in the boundary bounds a disk (which is rarely null-homologous) and ( p CP q CP ) ◦ , in which we find plenty of null-homologous disks (but onlyknow how many summands p, q we need after fixing a knot on the boundary) a natural questionconcerns the existence of a universal slicing manifold. Is there a fixed compact, smooth, oriented4-manifold V with ∂V = S such that any knot K ⊂ S is slice in V via a null-homologousdisk? It turns out that a signature estimate shows such a universal solution cannot exist. Theorem 4.4.
Any compact oriented -manifold V with ∂V = S contains a knot in itsboundary which is not topologically null-homologous slice in V . Remark 4.5.
If we drop the assumption that V should be compact, a punctured infiniteconnected sum of projective planes does the job: D ∞ ( CP CP ) For any fixed knot on the boundary there is a compact slice disk in a finite stage D k CP l CP D ⊂ D ∞ ( CP CP ) . The remainder of this section is concerned with a proof of Theorem 4.4. As preparation,let us specialize a result [CN20, Thm. 3.8], which is a generalization of the Murasugi-Tristraminequality for links bounding surfaces in 4-manifolds, to the case of knots. Here σ ω ( K ) is the Levine-Tristram signature of the knot K , defined as the signature of the hermitian matrix(1 − ω ) V + (1 − ω ) V T , where V is a Seifert matrix of K and ω a unit complex number not equalto 1. References for this signature include [Lev69], [Tri69] and the recent survey [Con19]. Thefollowing inequality only holds for specific values of ω , and will adopt the notation S for unitcomplex numbers ω ∈ S − { } which do not appear as a zero of an integral Laurent polynomial p ∈ Z [ t, t − ] with p (1) = ± Theorem 4.6 ([CN20, Special case of Thm. 3.8]) . Let X be a closed oriented topological -manifold with H ( X ; Z ) = 0 . If Σ ⊂ ( S × I ) X is a null-homologous (topological) cobordismbetween two knots K , K , each living respectively in one of the two boundary component S ’s of ( S × I ) X , then | σ K ( ω ) − σ K ( ω ) + sign( X ) | − χ ( X ) + 2 ≤ − χ (Σ) for all ω ∈ S . For K ⊂ ∂X ◦ which is null-homologous slice in X we can further simplify: Corollary 4.7.
Let X be a closed topological -manifold with H ( X ; Z ) = 0 . If the knot K ⊂ S is topologically null-homologous slice in X ◦ then for ω ∈ S we have | σ K ( ω ) + sign( X ) | − χ ( X ) + 2 ≤ H , then pick the knot K in the originalmanifold boundary and arrive at a contradiction to Corollary 4.7 in the surgered manifold if K was null-homologous slice. Proof of Theorem 4.4.
Let V be a compact topological 4-manifold with boundary S , we want tofind a knot in its boundary which is not slice. Pick a set of disjointly embedded loops γ , . . . , γ l in V whose homology classes generate H ( V ). If V already satisfies H ( V ) = 0, set l = 0 for theremainder of the proof and omit the surgery altogether. Let K be a knot in S whose signature(at the unit complex number ω = −
1) satisfies | σ K ( − | ≥ | sign( V ) | + | χ ( V ) | + 2 l. Note that the constant on the right hand side only depends on the signature, Euler characteristic,and number of generators of H ( V ), and not on the knot K . For example, since signature isadditive under connected sum, the self-sum K n = n K with n large enough has arbitrarily highsignature at ω = − K that has positive signature σ K ( − K is slice in V via a null-homologous disk ∆. Being null-homologous in therelative second homology group means geometrically that there is a locally flat embedded3-manifold M with boundary the slice disk ∆ union a Seifert surface for K in the boundary S , see [Lic97, Lem. 8.14]. We can remove the closed components from M , what remains isa 3-manifold with nonempty boundary in V . Generically the embedded circles γ , . . . , γ l willintersect the 3-manifold M in points, but we can push these intersection points off the boundaryof M via an isotopy of the curves in V . We will still keep the notation γ , . . . , γ l for the isotopedcurves which are now disjoint from M . Essentially, this finger move supported in a neighborhoodof M is guided by pairwise disjoint arcs in M connecting the intersections points to the boundary.Perform surgery on the loops γ , . . . , γ l , i.e. for each γ i remove an open tubular neighborhood ν ( γ i ) ∼ = S × int D and glue copies of D × S to the new S × S boundary components viathe identity map S × S → S × S . After this surgery we have a compact 4-manifold V EEP AND SHALLOW SLICE KNOTS IN 4-MANIFOLDS 11 with H ( V ) = 0, and the original disk ∆ survives into V in which we will call it ∆ . Observethat this “new” disk ∆ is still null-homologous in V , since the 3-manifold is still present afterthe surgery. Each circle surgery in a 4-manifold increases the Euler characteristic by 2, thus χ ( V ) = χ ( V ) + 2 l . By construction, the 4-manifolds V and V are cobordant, and so theirsignatures sign( V ) = sign( V ) agree.Starting with a knot K with high enough signature, if there existed a null-homologous ∆ ,since H ( V ) = 0: (cid:12)(cid:12) σ K ( −
1) + sign( V ) (cid:12)(cid:12) − χ ( V ) + 2 = | σ K ( −
1) + sign( V ) | − ( | χ ( V ) | + 2 l ) + 2 > K cannot exist. (cid:3) Remark 4.8.
Earlier sources for results in the smooth category include Gilmer and Viro’s[Gil81] version of the Murasugi-Tristram inequality for the classical signature as stated in [Yas96,Thm. 3.1]. Our preference for using [CN20] in the proof of Theorem 4.4 comes from the resultbeing stated in the topological locally flat category.5.
Speculation and Questions
Connection to other conjectures.
An alternative approach to the SPC4 is to find acompact 3-manifold M that embeds smoothly in some homotopy 4-sphere Σ , but not in S .Notice that if a smooth integral homology sphere M smoothly embeds in Σ, then M is theboundary of a smooth homology 4-ball [AGL17, Prop. 2.4]. However, there is no known exampleof a 3-manifold M that is the boundary of a smooth homology 4-ball but that does not embedinto S . Both this and the approach in the introduction are hung up at the homological level.Further discussion of knots in homology spheres and concordance in homology cylinders can befound in, for example, [HLL18], [Dav19].Corollary 2.5 has some relevance to this which we now discuss (a similar discussion also appearsin a comment by Ian Agol on Danny Calegari’s blogpost [Ago13]). The unsolved Schoenfliesconjecture proposes that if S ⊂ S is a smoothly embedded submanifold with S homeomorphicto S , then S bounds a submanifold B ⊂ S that is diffeomorphic to D . The SPC4 implies theSchoenflies conjecture. Question 5.1.
Does every exotic homotopy 4-ball B smoothly embed into S ?Note that if the answer to Question 5.1 is yes, then the Schoenflies conjecture implies theSPC4 and hence the two conjectures are equivalent: If any homotopy 4-ball would embed into S and thus, by the Schoenflies conjecture, would be diffeomorphic to D , hence all homotopy4-balls would be standard, so all homotopy 4-spheres would be standard. We have: Observation 5.2.
If the answer to Question 5.1 is yes, then no homotopy 4-ball can have deepslice knots.
Thus by Corollary 2.5, if the answer to Question 5.1 is yes, the approach towards SPC4mentioned in this section would never succeed. Similarly, there would be no 3-manifold thatwould smoothly embed into a homotopy 4-sphere but not into S , so this approach to SPC4would also be a dead end.5.2. More questions.Question 5.3.
Are there any 2-handlebodies X other that \ k ( S × B ) , k ≥
0, with the propertythat all K ⊂ B ⊂ X that are slice in X are also slice in B ? In other words, are there alwaysdeep slice local knots when X = \ k ( S × B )?One strategy for answering this question would be to start with a framed link L describing a2-handlebody other than \ k ( S × B ) and to handle-slide L to a new framed link L that containsa knot K that is not slice in B . Then this knot K when considered in a 3-ball K ⊂ B ⊂ ∂X is an example of such a deep slice knot in X . This strategy fails to find any non-slice knots K (as it must) for \ k ( S × D ) when we start with L being the 0-framed unlink – since then allresulting knots K will be ribbon hence slice in B .In view of the 2-handlebodies constructed in Proposition 4.1, one could ask whether thisextension of the Norman-Suzuki trick is the only way to make any knot in the boundary of amanifold bound an embedded disk: Question 5.4. If X is a 2-handlebody with the property that every knot in the boundary of X is slice in X (no assumption on the relative homology class of the disk), does it follow that X decomposes as X = X S × S ) or X = X S e × S )? More generally, what about the samequestion without the hypothesis that X be a 2-handlebody? References [AGL17] Paolo Aceto, Marco Golla, and Kyle Larson. Embedding 3-manifolds in spin 4-manifolds.
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Email address : [email protected] URL : https://math.berkeley.edu/~mrklug/ Max-Planck-Institut für Mathematik, Bonn, Germany
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