In this paper we exhibit infinite families of embedded tori in 4-manifolds
that are topologically isotopic but smoothly distinct. The interesting thing
about these tori is that they are topologically trivial in the sense that each
bounds a topologically embedded solid handlebody. This implies that there are
stably ribbon surfaces in 4-manifolds that are not ribbon.Comment: 8 pages, 2 figure, remarks added explaining the relevance of this
  work to stably ribbon surface

Hoffman, Neil

Sunukjian, Nathan

English

arXiv

We exhibit infinite families of embedded tori in 4–manifolds that are topologically isotopic but smoothly distinct. The interesting thing about these tori is that they are topologically trivial in the sense that each bounds a topologically embedded solid handlebody. This implies that there are stably ribbon surfaces in 4–manifolds that are not ribbon

Hoffman, Neil R.

Calvin Digital Commons (Calvin University)

Calvin University Calvin Digital Commons University Faculty Publications and Creative Works University Faculty Scholarship 1-1-2020 Null-homologous exotic surfaces in 4–manifolds Nathan Sunukjian Calvin University Neil R. Hoffman Oaklahoma State University Follow this and additional works at: https://digitalcommons.calvin.edu/calvin_facultypubs  Part of the Applied Mathematics Commons Recommended Citation Sunukjian, Nathan and Hoffman, Neil R., "Null-homologous exotic surfaces in 4–manifolds" (2020). University Faculty Publications and Creative Works. 132. https://digitalcommons.calvin.edu/calvin_facultypubs/132 This Article is brought to you for free and open access by the University Faculty Scholarship at Calvin Digital Commons. It has been accepted for inclusion in University Faculty Publications and Creative Works by an authorized administrator of Calvin Digital Commons. For more information, please contact digitalcommons@calvin.edu. NULL-HOMOLOGOUS EXOTIC SURFACES IN 4-MANIFOLDSNEIL R. HOFFMAN AND NATHAN S. SUNUKJIANAbstract. In this paper we exhibit infinite families of embedded tori in 4-manifolds that are topologically isotopic but smoothly distinct. The interestingthing about these tori is that they are topologically trivial in the sense thateach bounds a topologically embedded solid handlebody. This implies thatthere are stably ribbon surfaces in 4-manifolds that are not ribbon.1. introductionJust as a 4-manifold can have many inequivalent smooth structures, there canbe many different smooth embeddings of surfaces into a 4-manifold which are topo-logically isotopic, but smoothly distinct. Any surface which admits more thanone smooth embedding in a topological isotopy class will be said to admit exoticembeddings.In this paper we will show that null-homologous tori first discovered by Fintusheland Stern in their knot surgery construction in fact provide examples of exoticallyembedded tori. Specifically,Theorem 1.1. Let X be a smooth 4-manifold with b2 ≥ |σ|+6, non-trivial Seiberg-Witten invariant, and embedded torus T of self intersection 0 such that π1(X \T ) =1. Then X contains an infinite family of distinct tori {Ti} that are topologicallyisotopic to the unknotted torus (a torus that bounds a solid handlebody in X), butno diffeomorphism of X exists taking Ti to Tj if i 6= j.The first examples of orientable exotic embeddings come from Fintushel andStern’s “rim surgery” technique [4]. Their surfaces all have simply connected com-plement. A variation on rim surgery was given by Kim, and Kim-Ruberman whichworks in the case that the complement has non-trivial fundamental group ([10–12]).Tom Mark has used Heegaard-Floer homology to show that these constructions arealso effective for constructing exotic embeddings of surfaces with negative self inter-section ([13]). On the other hand, all of these constructions involve surfaces whosecomplement has finite first homology, and moreover all of these constructions es-sentially begin with symplectically embedded surfaces in a symplectic 4-manifold.Such surfaces can never be null-homologous. The significance of our examples isthat they are null-homologous and moreover topologically trivial.1arXiv:1310.1838v3  [math.GT]  1 Aug 20192 NEIL R. HOFFMAN AND NATHAN S. SUNUKJIANOne of the features of this theorem is that there are numerous tractable exampleswhere the theorem can be applied. For example, any elliptic surface contains sucha torus and has non-trivial Seiberg-Witten invariant by virtue of being a symplecticmanifold.The strategy of proof is as follows: The knot surgery construction of Fintusheland Stern produces an infinite family of exotic smooth structures on a 4-manifoldthrough a series of log-transforms on null-homologous tori. These are the toriwe will focus on. Using Seiberg-Witten theory, we will define a gauge theoreticinvariant of null-homologous tori to distinguish the tori smoothly. Finally, we willshow that all such tori are topologically isotopic by a theorem of the second author:Theorem 1.2 ([15, Theorem 7.2]). Let Σ0 and Σ1 be locally flat embedded sur-faces of the same genus in a simply connected 4-manifold X. The surfaces aretopologically isotopic when π1(X \ Σi) = Z and b2 ≥ |σ|+ 6.Note that a trivially embedded surface in any 4-manifolds will satisfy π1(X\Σ) =Z.One might wonder how robust these exotic embeddings are. That is, whatdoes it take to make any of the exotically embedded topologically trivial surfacesconstructed here smoothly equivalent again? In [1], Inanc Baykur and the secondauthor show that these tori become smoothly equivalent once one increases thegenus of each of these surfaces in the most trivial possible way. Namely, tubing anyone of the topologically trivial surfaces of Theorem 1.1 to a smoothly trivial torusresults in a smoothly trivial surface.It would be interesting to know what the simplest examples of exotic embed-dings are. For example, this paper provides context for the following two naturalquestions, which have motivated work on exotically embedded surfaces.Question. Do there exist exotically embedded surfaces in S4? In particular, isthere an embedded S2 that is topologically isotopic to the unknot but not smoothlyisotopic to the unknot?The examples in this paper can be seen as prototypes for answering this sort ofquestion, as answering this question in other manifolds can be seen as a first stepto better understanding the S4 question. At the same time, the examples of thispaper can better inform an attack on the question above.We conclude the introduction with the following remark:Remark 1. The exotically embedded tori constructed in this paper provide examplesof stably ribbon surfaces that are not ribbon. They are stably ribbon, because theystably smoothly trivial (by [1]). But they are not ribbon because as we shall see, logtransforms on these tori have a different effect than log transforms on trivial tori,NULL-HOMOLOGOUS EXOTIC SURFACES IN 4-MANIFOLDS 3whereas [15, Theorem 8.3] implies that log transforms on ribbon tori are equivalentto those on trivial tori.Acknowledgements: Both authors would like to thank the Max Planck In-stitute for Mathematics for hosting them while they worked on this project, andDanny Ruberman and Tom Mark for their comments on an early draft of thispaper. We are also especially grateful to Inanc Baykur who provided extensivecomments on a later version. The first author was partially supported by grantfrom the Simons Foundation (#524123 to Neil R. Hoffman) during later revisionsof this paper.2. Constructing the toriLet T be an embedded torus with self intersection zero in a 4-manifold X suchthat π1(X\T ) = 1. We will not construct exotic embeddings of T , (any such torus isnecessarily homologically essential since it will have a dual), but rather we will findexotic embeddings of nearby null-homologous tori which arise in the “knot surgery”construction of Fintushel and Stern ([3] and [5]). Knot surgery along torus T using aknot K ⊂ S3 is most straightforwardly defined as XK = (X\ν(T ))∪(S1×S3\ν(K))where the union is formed by taking the longitude of K to the meridian of T (apartfrom this requirement, the gluing is not, strictly speaking, well defined, and XK maydepend on the gluing map, but this ambiguity does not factor into our argumentbelow). Fintushel and Stern proved that X is homeomorphic to XK under theassumption that the complement of T is simply connected, and they further provedthat their Seiberg-Witten invariants are related by SWXK = SWX ·∆K(2[T ]) where∆K is the Alexander polynomial for K. Therefore, by varying K, one can constructinfinitely many smooth structures on X.The Seiberg-Witten formula is proved by viewing knot surgery as a series oflog-transforms on null-homologous tori. That is, rather than cutting out ν(T ) =S1× (S1×D2) and replacing it with S1×S3 \ν(K), we can view knot surgery as aseries of log-transforms in S1× (S1×D2) which eventually lead to S1×S3 \ ν(K).Forgetting the extra S1 direction for the moment, one can go from S3 \ ν(K) to(S1×D2), the complement of the unknot, by doing ±1 surgery along crossings of Kto unknot it. Crossing this whole picture with S1 gives the log-transforms neededfor knot surgery. It is these null-homologous tori that are needed to do the knotsurgery that will turn out to be the exotically unknotted tori that we are lookingfor in our theorem.We’ll focus on the following particular situation: Suppose that K is a knotof unknotting number 1, and T is a torus in X with trivial normal bundle andsimply connected complement. Then knot surgery is the result of doing a singlelog transform on the null homologous torus TK = S1 × γK in νT ⊂ X (see Figure4 NEIL R. HOFFMAN AND NATHAN S. SUNUKJIANknotsurgery2.{ps,eps,pdf} not found (or no BBox)Figure 1. The left figure represents νT , which can be thought ofas the complement of the unknot in S3 crossed with S1. Performinga (+1)-log transform on TK = S1×γK gives S1×(S3\νK), depictedon the right.1). The proof of our theorem will show that TK is topologically unknotted, butsmoothly non-trivial. To determine the topological isotopy class of TK we will nowcompute the fundamental group of X \ TK : Note that since π1(X \ T ) = 1, wehave that the inclusion π1(∂νT ) → π1(X \ νT ) is trivial. Therefore, by repeatedapplications of Van-Kampen’s theorem,π1(X \ TK) = π1((X \ νT ) ∪ (νT \ TK)=π1(νT \ TK)π1(∂νT )=π1(S1 × (S3 \ (νU ∪ γK)))〈S1〉 × π1(∂νU)= π1(S3 \ γk) = ZThe second to last line is just a change of notation, since νT \ TK is just thesame thing as S1 × (S3 \ (νU ∪ γK)) where U is the unknot (see Figure 1). Andthe final equality is true because γk is necessarily unknotted in S3.Already we see that this gives at least one exotically embedded torus. Specif-ically, TK is topologically standard by Theorem 1.2, and moreover, performing alog-transform on TK will give an exotic smooth structure on X, whereas performinga log-transform on the standardly embedded torus, (i.e. the one that bounds a solidhandlebody), will not. Therefore these tori are smoothly distinct, but by Theorem1.2 they must be topologically isotopic.To construct infinite families of exotic surfaces, we need to be more careful. Forexample suppose K1 and K2 are knots with associated null homologous tori TK1and TK2 . Then it is conceivable that one might be able to construct both XK1and XK2 by some surgery on TK1 . In this circumstance we would not be able todistinguish TK1 from TK2 as we did above. To resolve this issue, we have to lookmore deeply at how the Seiberg-Witten invariant changes under log-transforms onTK , and restrict ourselves to certain classes of knots.3. Smooth invariants of null-homologous toriThe Seiberg-Witten invariant of a 4-manifold X is a map SWX : S −→ Z, whereS is the set of isomorphism classes of spinc structures on X. The basic classes ofX are defined to be the spinc structures that map to non-zero integers. It is a wellknown property of the Seiberg-Witten invariant that a closed 4-manifold has only aNULL-HOMOLOGOUS EXOTIC SURFACES IN 4-MANIFOLDS 5finite number of basic classes. Below, we will often not distinguish between a spincstructure and its first Chern class or even the Poincare dual of its first Chern class.We will distinguish our null-homologous tori by computing an invariant that is,in a technical sense clarified below, related to the Seiberg-Witten basic classes ofthe complement of the tori. To do this we will need to understand how the Seiberg-Witten invariant of a 4-manifold is affected by log-transforms. Suppose we aregiven a 4-manifold with T 3 boundary, e.g. X \νT , and suppose H1(T 3) = Z[a, b, c].Denote the log-transformed 4-manifold constructed by gluing on a D2 × T 2, where[∂D2] is glued to [pa+ qb+ rc] as XT (p, q, r) or sometimes just X(p,q,r), and denotethe core torus in the D2 × T 2 part of this manifold as T(p,q,r).A formula of Morgan-Mrowka-Szabo from [14] give a formula relating the Seiberg-Witten invariants of various log-transforms:∑iSWXT (p,q,r)(k(p,q,r) + i[T(p,q,r)]) = p∑iSWXT (1,0,0)(k(1,0,0) + i[T(1,0,0)])(1)+ q∑iSWXT (0,1,0)(k(0,1,0) + i[T(0,1,0)]) + r∑iSWXT (0,0,1)(k(0,0,1) + i[T(0,0,1)])where the spinc structures agree away from the log-transformed tori, i.e.:k(p,q,r)|X(p,q,r)\T(p,q,r) = k(1,0,0)|X(1,0,0)\T(1,0,0)= k(0,1,0)|X(0,1,0)\T(0,1,0)(2)= k(0,0,1)|X(0,0,1)\T(0,0,1) .Suppose that T(p,q,r) is null-homologous. Then the left hand side of Equation 1has only one term. Moreover, since X(1,0,0) has only a finite number of basic classesk(1,0,0) + i[T(1,0,0)], we see that Equation 2 implies that there are only a fixed finitenumber of possible basic classes k(p,q,r) for XT (p,q,r), and these possibilities dependonly on the choice of T , not on (p, q, r). To put this another way, there are only afinite number of spinc structures on X \ νT that can be extended to basic classeson XT (p, q, r) when [T(p,q,r)] = 0. Therefore, the following invariant is well defined:Definition. Let T be a null-homologous torus in X (defined up to smooth isotopy).Define B(X,T ) to be the maximum divisibility of the difference between any twobasic classes of XT (p, q, r) for any (p, q, r) such that [T(p,q,r)] = 0.Remark 2. It is important to use the divisibility of the difference of basic classesrather than just the divisibility of the basic classes because very often after per-forming knot surgery all of the basic classes have divisibility 1.6 NEIL R. HOFFMAN AND NATHAN S. SUNUKJIANa0a1a2a3 a2n−1. . .Figure 2. The two-bridge knot C(a0, . . . , a2n−1).4. Families of unknotting number one knots, and the proof ofTheorem 1Now that we have a better understanding of the smooth invariants needed todistinguish potential infinite families of smooth tori, we can describe an explicitfamily of knots that will give rise to smoothly distinct TK . For the invariant B(X,T )to be useful, we will need to find a family of knots with unknotting number 1 whoseAlexander polynomials have arbitrarily high degree. All two-bridge knots can begiven in the form of Figure 2 where ai is the number of right half-twists wheni is odd, and left half-twists when i is even. We refer to two-bridge knots usingConway’s notation, C(a0, . . . , am), and we note that it is well known (see [2, §12B]for instance), that two 2-bridge knots are equivalent if and only if [a0, . . . , am] and[a′0, . . . , a′m′ ] are continued fraction expansions of the same rational number.Proposition 4.1 (Kanenobu-Murakami [9]). A two-bridge knot has unknottingnumber one if and only if it can be expressed asC(b, b1, b2, . . . , bk,±2,−bk, . . . ,−b2,−b1).The following proposition of Burde-Zieschang tells us how to compute the rele-vant polynomial invariants:Proposition 4.2 (Burde-Zieschang [2, Proposition 12.23]). The Conway polyno-mial of a two-bridge knot expressed as C(a0, . . . , a2n−1) has degree∑n−1j=0 |a2i|.Remark 3. The notation in Burde-Zieschang is different than that used in Proposi-tion 4.1. To reconcile the conventions, the two-bridge knot diagram in Figure 2 canbe converted to a 4-plat diagram as in Burde-Zieschang by pulling the inner strandon the right hand side of the figure over the outer strand. This has the effect ofadding a new crossing (i.e. a2n = +1) and adjusting a2n−1 by +1.Lemma 4.3. There exists an infinite family of unknotting number one knots whoseAlexander polynomials have arbitrarily high degree.Proof. Combining Propositions 4.1 and 4.2 shows that there exists an infinite familyof two-bridge knots of unknotting number one such that the Conway polynomial hasarbitrarily high degree. The lemma is thus immediate from the fact that the ConwayNULL-HOMOLOGOUS EXOTIC SURFACES IN 4-MANIFOLDS 7polynomial is related to the Alexander polynomial by the formula ∇(t − t−1) =∆(t2). Proof of Theorem 1.1. Let {Kj} be a sequence of knots of unknotting number 1such that the degree of their Alexander polynomials goes to infinity, and let TKjbe the associated (topologically trivial) tori from Section 2.Since there is a log-transform on TKj that gives XKj , we have thatlimj→∞B(X, TKj ) ≥ limj→∞(max divisibility of the differencebetween any two basic classes of XKj)≥ limj→∞4deg(∆Kj ) =∞.The second inequality follows because the knot surgery formula,SWXK = SWX ·∆K(2[T ]),allows us to determine the basic classes of XK from those of X. Specifically, if κ isa basic class of X, and the degree of ∆Kj is n, then XKj has κ+2n[T ] and κ−2n[T ]as basic classes (among others), and the divisibility of the difference of this pair ofbasic classes is 4n, which serves a a lower bound for B(X, TKj ). Therefore, afterpossibly passing to a subsequence of the {XKj} there are an infinite number of theTKj that are smoothly distinguished by their B invariant. References[1] Inanç Baykur and Nathan Sunukjian, Knotted surfaces in 4-manifolds and stabilizations, Jour-nal of Topology 9 (2015), 215-231.[2] Gerhard Burde and Heiner Zieschang, Knots, 2nd ed., de Gruyter Studies in Mathematics,vol. 5, Walter de Gruyter & Co., Berlin, 2003.[3] Ronald Fintushel and Ronald J. Stern, Knots, links, and 4-manifolds, Invent. Math. 134(1998), no. 2, 363–400.[4] , Surfaces in 4-manifolds, Math. Res. Lett. 4 (1997), no. 6, 907–914.[5] Ronald Fintushel, Knot surgery revisited, Floer homology, gauge theory, and low-dimensionaltopology, Clay Math. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 2006, pp. 195–224.[6] Ronald Fintushel and Ronald J. Stern, Surgery on nullhomologous tori and simply connected4-manifolds with b+ = 1, J. Topol. 1 (2008), no. 1, 1–15.[7] Ian Hambleton and Peter Teichner, A non-extended Hermitian form over Z[Z], ManuscriptaMath. 93 (1997), no. 4, 435–442.[8] Stefan Friedl, Ian Hambleton, Paul Melvin, and Peter Teichner, Non-smoothable four-manifolds with infinite cyclic fundamental group, Int. Math. Res. Not. IMRN 11 (2007), Art.ID rnm031, 20.[9] Taizo Kanenobu and Hitoshi Murakami, Two-bridge knots with unknotting number one, Proc.Amer. Math. Soc. 98 (1986), no. 3, 499–502.[10] Hee Jung Kim, Modifying surfaces in 4-manifolds by twist spinning, Geom. Topol. 10 (2006),27–56 (electronic).[11] Hee Jung Kim and Daniel Ruberman, Topological triviality of smoothly knotted surfaces in4-manifolds, Trans. Amer. Math. Soc. 360 (2008), no. 11, 5869–5881.[12] , Smooth surfaces with non-simply-connected complements, Algebr. Geom. Topol. 8(2008), no. 4, 2263–2287.[13] Thomas Mark, Knotted surfaces in 4-manifolds, Forum Mathematicum 25 (2013), no. 3,597–637.8 NEIL R. HOFFMAN AND NATHAN S. SUNUKJIAN[14] John W. Morgan, Tomasz S. Mrowka, and Zoltán Szabó, Product formulas along T 3 forSeiberg-Witten invariants, Math. Res. Lett. 4 (1997), no. 6, 915–929.[15] Nathan S Sunukjian, Surfaces in 4-Manifolds: Concordance, Isotopy, and Surgery, Int. Math.Res. Not. IMRN 17 (2015), 7950–7978.Department of MathematicsOklahoma State University401 Mathematical Sciences Building,Stillwater, OK 74078-1058E-mail address: neil.r.hoffman@okstate.eduDepartment of Mathematics and StatisticsNorth HallCalvin College3201 Burton SEGrand Rapids, MI 49546E-mail address: nss9@calvin.edu

Null-homologous exotic surfaces in 4-manifolds

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