Josef G.DORFMEISTER
Department of Mathematics,North Dakota State University,Fargo,ND 58102,USA
E-mail:josef.dorfmeister@ndsu.edu
Tian-Jun LI†
School of Mathematics,University of Minnesota,Minneapolis,MN 55455,USA
E-mail:tjli@math.umn.edu
Abstract This paper gives a survey of the progress on the minimal genus problem since Lawson’s [26]survey.
Key words four manifolds;minimal genus;adjunction-type inequality;circle sum
In honor of Professor Banghe Li,on the occasion of his 80’th birthday,in recognition of his fundamental contributions to the minimal genus problem.
LetMbe a smooth,closed 4-manifold.Then,for any given classA∈H2(M,Z),there exists a connected,smoothly embedded surfaceΣwith [Σ]=A.It is possible to increase the genus of Σ to obtain a homologous surface.On the other hand,determining the smallest possible genus of such a surface representingA,the minimal genusgm(A),is generally very hard.With the advent of gauge theoretic methods in 4-manifold theory in the 80’s and 90’s it became possible to determine this genus for certain manifolds.These results are described in the excellent survey by Lawson [26].
In the quarter century that has passed since that survey was written,further progress has been made on the minimal genus problem.Some of these resultscontinue to use gauge theoretic methods;other results begin to answer the question for classesfor which Seiberg-Witten theory fails to provide any information.
In general,this problem has been addressed as follows:
(1)Submanifolds with an auxiliary structure,for examplea complex structureor symplectic structure,are studied,and a generalized version of the Thom conjecture is proven for these structures.Such submanifolds are known to be genus minimizing.
(2)A lower bound on the genus is determined using an adjunction type inequality.An upper bound is determined through an explicit construction of an embedded,connected surface.
Adjunction-type inequalities,leading to a lower bound on the possible minimal genus,are surveyed in Section 2.Wall [49]showed that there cannot be any cohomology adjunction formulas for manifolds with strongly indefinite intersection forms(b±≥2).It thus remains to study the case of manifolds withb±=1,or those with definite intersection forms.Section 2.4 surveys results forb+=1,largely for surfaces of non-negative self-intersection(or,equivalentlyb−=1 and non-positive self-intersection);see also the introduction of [2].
The remainder of this survey describes minimal genus determinations obtained for certain classes of manifolds:rational manifolds,ruled manifolds and elliptic surfaces.A new result for pairs of disjoint surfaces inb±=2 definite manifolds is given in Section 3.4.
This survey focuses on orientable surfaces in closed manifolds.Levine-Ruberman-Strle [29]studied non-orientablesurfacesΣof genusgin four-manifoldsM.The main result hereconcerns a homology cobordism between rational homology spheres,and provesan inequality on the genus of an essential embeddingΣ.The statement(and proof)of this result involvescertain invariants defined by Osv´ath-Szab´o [45].They also obtained adjunction-type results adapted to the nonorientable setting for certain four-manifolds;see also Iida-Mukherjee-Taniguchi(Theorem 4.6,[15])for an adjunction-type result concerning manifolds with a boundary.Notably,Gompf([12])produces new powerful existence results for exotic smooth structures on open manifolds using a minimal-genus function and its analogue for end homology.
Notation and ConventionsH2(M,Z)andH2(M,Z)will beidentified via Poincar´e duality and will not be distinguished unless crucial.A classc∈H2(M,Z)is called characteristic ifc·A=A·A(mod 2)for allA∈H2(M,Z)andσ(M)denotes the signature of the manifoldM.Throughout,unless otherwise stated,homology and cohomology are assumed to be torsion free.
In this section we describe some adjunction-type inequalities obtained for certain classes of manifolds;most of these result from Seiberg-Witten theory or perturbations thereof.
Definition 2.1LetA∈H2(M,Z)and let Σ be a connected embedded surface representing the classA.Letχ(Σ)denote the Euler characteristic of this surface.An adjunction-type inequality is an inequality of the form
whereA·Adenotes the self-intersection given by the intersection pairing ofM,andf(A)is a function generally of the following structure:for the manifoldM,a special set of classes inH2(M,Z)is determined and evaluated,using the intersection pairing,againstA,sof(A)is a degree 1 function ofAin some way optimal with respect to this set of values.
2.1 Ozsv´ath-Szab´o’s Adjunction Inequalities
The original Thom conjecture asserts that the genus of an embedded surface Σ representing a homology classA∈H2(CP2,Z)is minimized by an algebraic curve.This conjecture has been proven and subsequently generalized in a number of different directions;see Theorems 12 and 29,[26].Ozsv´ath and Sz´aboproved a symplectic version as follows:
Theorem 2.2(Theorem 1.1,[43](Symplectic Thom Conjecture))An embedded symplectic surface in a closed,symplectic four-manifold is genus-minimizing in its homology class.
Previous results in this direction had only shown this to be true for those classes with nonnegative self-intersection.The key argument in their proof is a calculation of certain Seiberg-Witten relations for negative self-intersection surfaces(Thm 1.3,[43]).As an auxiliary result,they obtained an adjunction inequality for negative self-intersection classes(see Theorems 11 and 17,[26]for the non-negative case).
Theorem 2.3(Corollary 1.7,[43])LetMbe a smooth,closed four-manifold of simple type withb+>1 andΣa smoothly embedded,oriented,closed surface of genusg>0 and [Σ]·[Σ]<0.Then,for all Seiberg-Witten basic classess,
As is generally the case in Seiberg-Witten theory,ifb+=1,then this result holds under consideration of the chamber structure;see Remark 1.10,[43].
Similar techniques are applied in [44]to refi ne the adjunction inequality for non-negative self-intersection surfaces.
2.2 Lambert-Cole’s Ad junction Inequalities
In a series of papers,Lambert-Cole provided a proof of the Thom conjecture [22]and the symplectic Thom conjecture [23]making no use of gauge theory or any pseudoholomorphic techniques.Instead,the proof used trisections introduced by Gay and Kirby.
Furthermore,in [23]and [24],a variety of adjunction inequalities were proven which hold without any restrictions onA·A,the genus of the curve orb+.The following is one such example:
Theorem 2.4(Theorem 1.2,[23])Let(M,ω)be a close,symplectic four-manifold with [ω]integral.If Σ is a smoothly embedded surface with [Σ]·[ω]>0,then
This Theorem can be viewed as an extension of Theorem E,[38],in theb+=1 case to classes of negative self-intersection.
As noted by Lambert-Cole,“there is hope that extensions of [the techniques used in the proof]may yield genus information in smooth 4-manifolds with trivial Seiberg-Witten invariants,such as connected sums.”
2.3 Symplectic Genus
The Symplectic Thom Conjecture illustrates the role of symplectic surfaces in the minimal genus problem.It thus seems natural that any class which satisfies the homological necessary restrictions to be represented by a symplectic surface should allow for some restrictions on the possible genus arising from the symplectic structures onM.The symplectic genus defined by B-H.Li and T-J.Li(Definition 3.1,[33])offers such a bound.
Furthermore,Theorem 2.3 requires knowledge of Seiberg-Witten basic classes.In the symplectic setting,Taubes showed(see Theorem 6,[26])that the symplectic canonical classKωis a SW-basic class for any symplectic formω.
For a classA∈H2(M,Z)of a symplectic manifold(M,ω)to be represented by a symplectic surface,it must hold that [ω]·A>0.In other words,there must existα∈CMin the symplecticcone withα·A>0.Let K denotes the set of symplectic canonical classes and CM,K⊂CM,K∈K,the set of classes representable by symplectic formsωwithKω=K.Consider the following set:
and note thatη(A)=ηK(A)for anyK∈K(A).
Lemma 2.6(Lemma 3.2,[33])LetMbe a smooth,closed,oriented 4-manifold with non-empty symplecticcone.LetA∈H2(M,Z).Then the following hold:
(1)The symplectic genusη(A)is no larger than the minimal genus ofA.
(2)The symplectic genus is invariant under the action of Diff(M).
Notice that the first condition ensures that the symplectic genus is well-defined and that it also provides an obstruction to the existence of a smooth/symplectic surface.It also implies the genus inequality−χ(Σ)≥A·A+K·Afor any classK∈K(A).
Moreover,ifAis represented by a connected symplectic surface,then the minimal genus and the symplectic genus coincide.
Theorem 2.7(Theorem A,[33])LetMbe a smooth,closed,oriented four-manifold with a non-empty symplecticcone andb+=1.Then the symplectic genus of any class of positive squareisnon-negative and it coincides with the minimal genus for any sufficiently largemultiple of such a class.
Furthermore,it is shown that the bound given by the symplectic genus is sharp in a broad class of manifolds,in particular forM=andk≤9 andS2×S2(see Section 5,[26];Theorems 3.7 and 3.16,Lemma 3.9).
If the classes are allowed to have negative square,then it can be shown [3]that there exist examples which have negative symplectic genus.Hence,those classes cannot be represented by symplectic surfaces and the symplectic genus also does not appear to give any useful bound on the minimal genus.
2.4 Results for Manifolds with b+=1
The symplectic genus is defined for any symplectic four-manifold.However,the strongest results are obtained for manifolds withb+=1;see Theorem 2.7.It is thus natural to study results in this context.
2.4.1 Strle’s Results
Strle obtained the following adjunction-type inequality using Seiberg-Witten theory for manifolds with cylindrical ends:
Theorem 2.8(Theorem A,[47])LetMbe a smooth,closed,oriented four-manifold withb1=0 andb+=1.If Σ is a smoothly embedded surface of positive self-intersection,then
for any characteristic classc∈H2(M,Z)which satisfiesc·c>σ(M).
The bounds obtained by Strle depend only on the rational homology type of the manifold.In particular,the bounds can be shown to be sharp in CP2#kCP2andS2×S2,but examples areknown where the lower bound is not realized;see the Remarksfollowing Theorem 10.1,[47].
2.4.2 Cohomological Genus
Dai-Ho-Li defined the cohomological genus of a classAin purely cohomological terms and obtained lower bounds on the minimal genus of a surface ΣrepresentingApurely in terms of the cohomology ring of the 4-manifold.This generalizes Strle’s result.
For the purpose of this survey,we describe only the setup necessary to describe the result in the context of four-manifolds;algebraic details can be found in [2].
LetMbe a smooth,closed,connected,oriented four-manifold withb+=1.Consider the skew-symmetric bilinear form
(1)c·c>σ(M);
(2)c·c≥2˜χ(M)+3σ(M)andcpairs non-trivially with ImTwhenTis non-trivial.
Definition 2.9LetA∈H2(M,Z).For any adjunction classc,define
Define the cohomological genush(A)ofAbyh(A)=maxchc(A).
Dai-Ho-Li then obtained a result analogous to Lemma 2.6,which generalizes Strle’s Theorem 2.8.
Theorem 2.10(Theorem 1.4,[2])LetMbe a smooth,closed,connected,oriented fourmanifold withb+=1.Then,for anyA∈H2(M,Z)withA·A≥0,the following hold:
(1)The cohomological genush(A)is no larger than the minimal genus ofA.
(2)The cohomological genush(A)is invariant under the action of Diff(M).
This again implies the genus inequality−χ(Σ)≥A·A−|c·A|for any adjunction classc.Note that this does not involve any classes related to Seiberg-Witten invariants.
Theorem 2.11(Theorem 1.5,[2])LetMbe a smooth,closed,connected,oriented four manifold withb+=1 and assume that 2˜χ(M)+3σ(M)≥0.Thenh(A)≥0 for anyA·A>0 orA·A=0 andAis primitive.
It was also shown that the cohomological genus,under the conditionoffers a sharp bound for the minimal genus whenMis the connected sum of a manifoldYof Kodaira dimension−∞or 0 with an appropriate number of copies ofS1×S3.The list of possibleYis given explicitly in Theorem 1.5,[2].
2.5 Estimates for Configurations of Surfaces
Theorem 2.8 is only valid forb+=1,but the techniques used to prove it are readily able to prove a more general result for configurations ofnsurfaces in manifolds withb+=n.
Theorem 2.12(Theorem B,[47];Corollary 2.15,[19];[2])LetMbe a smooth,closed,connected four-manifold withb1=0 andb+=n>1.Let Σ1,···,Σnbe disjoint embedded surfaces inMwith positive self-intersections.Ifc∈H2(M,Z)is a characteristic class satisfying
then
holds for at least onei.
Konno(Theorem 2.11,[19])proved an analogous result for classes with vanishing self inter section.However,that result is only valid under a homological restriction(Condition 1,[19]).On the other hand,Konno was able to remove the condition thatb1=0.
In fact,the results of Strle,Konno and Dai-Ho-Li are very similar,although the techniques employed vary slightly.All employ Seiberg-Witten invariants:in Strle’s case for manifolds with cylindrical ends,in Konno the wall-crossing formulas are employed for a fixed spin c structure while two surfaces are considered,and in Dai-Ho-Li the surface is fixed and the two spin c structures are considered.
2.6 4-Manifolds Admitting a Circle Action
The study of the minimal genus problem for 4-manifolds admitting a circle action involves study of the corresponding problem for 3-manifolds.The complexity and Thurston norm defined for 3-manifolds below is very similar to the configuration results described in the previous section.For a survey of the minimal genus problem in 3-manifolds,see Kitayama [18]and Wu [51].
Definition 2.13 Let Σ be a compact surface with connected components Σ1,···,Σn.Then,
(1)define the complexity of Σ toχ−(Σ)=
(2)letA∈H2(Mk,Z)(dimMk=k)and define
Given a 3-manifoldN,the quantityx3(A)is known as the Thurston norm,although it only defines a seminorm onH1(N,Z)≡H2(N,Z),which can however be extended uniquely toH1(N,R).This can be viewed as a generalization of the knot genus;see Kitayama [18]for details.
Observe that the complexity may involve a configuration of surfaces,thus any bound on these quantities is related to the results of Strle and Konno,though it does not account for spheres.
A series of papers by Kronheimer([20,21]),Friedl-Vidussi [8]and Nagel [40]obtained the following result:
Theorem 2.14(Corollary 7.6,[20];Theorem 1.1,[8];Theorem 5.6,[40])LetNbe an irreducible,closed,oriented 3-manifold.Assume thatNis not a Seifert fibered space and is not covered by a torus bundle.Letp:M→Nbe an oriented circle bundle.Then each classA∈H2(M,Z)satisfies
The key to the proofs in [8]and [40]is to study the Seiberg-Witten invariants of an appropriate finite cover.In the case of graph manifolds,this necessitates finding a cover where the Turaev norm and the Thurston norm coincide,providing enough basic classes for estimates to work.
It is is also worth noting that this inequality is known to be sharp in cases where the class icadjunction inequality is not;see Section 4.1,[8].
2.7 Overview of Adjunction-Typ e Results
In this section,we endeavour to give an overview of the adjunction-type inequalities presented above and in Lawson’s survey.
2.7.1 Gauge Theory Invariants
LetMbe a smooth,closed four-manifold withb+≥2 orb−≥2.
(1)IfMadmits non-trivial gauge theory invariants(including symplectic manifolds),then an adjunction-type inequality holds:
Heref(A)is a degree 1 function depending on the gauge theory invariants;see Theorems 2.3 and 2.4.We also refer readers to Theorem 22,[9],which proves an adjunction inequality for oriented closed spin 4-manifold which has the same rational cohomology ring asK3#K3using the Bauer-Furuta stable cohomotopy SW invariant;see also Theorem 5.4,[10].
(2)IfMadmits no non-trivial gauge theory invariants,then,only weak adjunction inequalities
withα<1 exist.
•Characteristic Classes.Combining the results of Acosta(see Theorems 38 and 39,[26]),Hamilton [13]and Yasuhara(see Theorem 36,[26]),one obtains the following result:ifMis closed simply connected,A∈H2(M,Z)is characteristic,and eitherA·A<0(Hamilton)orA·Ais equivalent toσ(M)Pmod 16(Acosta,Yasuhara),then for any surfaceΣrepresentingA,we have a weak adjunction inequality withα=1/4.
•Divisible Classes.SupposethatA=2qB.Then the work of Rokhlin [46]producesa weak adjunction-type inequality withα=1/4.Bryan(see Theorem 43,[26],and also Kotschick-Matic,Theorem 42,[26])improved this slightly by assuming thatBis characteristic andqis small,for example whenq=1,α=15/16.
In contrast to these results,in a series of papers,Lee-Wilczynski determined the minimal genusfor topologically locally flat embeddings in simply connected four-manifolds;see Theorem 1.1,[27](a general estimate for arbitrary four-manifolds is given in Theorem 2.1,[28]).In the closed,simply connected case,this theorem can be stated as follows:
Theorem 2.15(Theorem 1.1,[27])LetMbe a closed,oriented,simply connected fourmanifold.Suppose thatA=dB∈H2(M,Z)is divisible with divisibilityd.Then there exists a simple,topologically locally flat embeddingΣrepresentingAby an oriented surface of genusg>0 if and only if
Ifb+=1,and eitherA·A≥0 ordis large enough,then this result leads to a weak adjunction-type inequality withα=1/2.Observe that this Theorem actually guarantees the existence of a surface if the inequality is satisfi ed.
•Primitive and Ordinary Classes.There exist manifolds where all such classes are represented by spheres;see Wall [49].
•Configurations of Disjoint Surfaces.An adjunction-type inequality withf(A)a degree 1 function,which is some cohomological evaluation onA,holds in certain settings for at least one of the surfaces;see Theorem 2.12.
2.7.2 b+=1,Classes with Non-Negative Self-Intersection
An adjunction-type inequality holds withf(A)a degree 1 function,which is some cohomological evaluation onA;see Theorems 2.8 and 2.10.
The results also hold ifb−=1 and considering classes with non-positive self-intersection,with appropriate absolute values.
2.7.3 Definite Manifolds with b±≥2
(1)Divisible or Characteristic Classes.The results in the strongly indefinite case all continue to hold and weak adjunction-type inequalities are known.
(2)Primitive and Ordinary Classes.The authors are only aware of a result for configurations ofb+disjoint surfaces;an adjunction-type inequality withf(A)a degree 1 function,which is some cohomological evaluation onA,holds in certain settings for at least one of the surfaces;see Theorem 2.12.
To determine,or bound from above,the minimal genusof a specific classusing an adjunction type inequality,it is necessary to explicitly construct a submanifold.Two key concepts are discussed in the following examples:
•There are a number of classical constructions such as the connected sum and smoothing of algebraic curves.A further method,the circle sum introduced in [36],is described below.
•Reducing the set of classes that need to be studied by using the induced action of the orientation preserving diffeomorphisms ofMonH2(M,Z).
3.1 Rational Manifolds
H2(M,Z)can be viewed from the following three different viewpoints with corresponding automorphism(sub)groups:
(1)The homology of the manifoldM.The geometric automorphism group is given by
(2)An integer latticeLwith quadratic formQ(the intersection form onM)together with the orthogonal groupO(L)of lattice automorphisms preservingQ.
(3)([52])The latticeH2(M,Z)is the root lattice of a Kac-Moody algebra,Qis the generalized Cartan matrix,and the Weyl groupWis the subgroup ofO(L)generated by reflections on classes withQ(x,x)∈{−1,−2}.A reflection ofBon the classAis given byrA(B)=B−Note that fork≤9,this is a hyperbolic Kac-Moody algebra.
ClearlyD(M)⊂O(L).In fact,the following is true:
Theorem 3.1([7,33,52])(1)Ifk≤9,thenW=D(M)=O(L).
(2)Ifk≥10,thenD(M)is a proper subgroup ofW.
Moreover,B.H.Li and T.J.Li [35]identified explicit generators ofD(M)(see also [48]and [52]).
Definition 3.2Two classesA,B∈H2(M,Z)are calledD(M)-equivalent if there is aσ∈D(M)such thatσ(A)=B.Denote byOAthe orbit of the classAunder the action ofD(M).
Homology classesin rational manifolds underD(M)-action exhibit two special classes.The first,reduced,arises as the elements of the fundamental chamberC;see [16]and [48].The second is simplified classes,which form the counterpart to the reduced classes;see [34].
Definition 3.3LetA∈H2(M,Z).
(1)A classA=(a,b1,···,bk)is called reduced ifa≥0,b1≥···≥bk≥0 and
(a)a≥b1(k=1),
(b)a≥b1+b2(k=2)or
(c)a≥b1+b2+b3(k≥3).
(2)A classA=(a,b1,···,bk)is called simplified ifa≥0,b1≥···≥bk≥0 and
(a)2a≤b1+b2(k=2)or
(b)3a≤b1+b2+b3(k≥3).
It is not hard to see that ifk≤9 andA2<0,thenAmay be equivalent to a simplified class,but not a reduced one.Fork≥10,A2<0,Amay be equivalent to either type.For each fixed valueA2=−n,the set of simplified classes is always finite.
Lemma 3.4([16,31,34,52])LetM=andA∈H2(M,Z)withA0.
(1)AisD(M)-equivalent to a reduced or simplified class.
(2)Each orbitOAcontains either a simplified class or a reduced class,never both.There is an efficient algorithm to find a representative in each orbit.
(3)IfAisD(M)-equivalent to a reduced class,then this is the unique reduced class inOA.
The case for simplified classes is slightly more subtle.
Lemma 3.5([4])LetM=and letA∈H2(M,Z)withA2=−n<0.Assume thatAis primitive and that theD(M)-orbitOAofAcontains a simplified class.Then,for each(k,n),n>0,the following is true:
(1)IfAis ordinary,thenOAis the unique primitive ordinary orbit.
(2)Ifn≡42 andAis characteristic,thenOAis the unique primitive characteristic orbit.
Fork=2 this is known to be false by an example of C.T.C.Wall(see [51]).
These reductions imply that to understand the minimal genus problem for rational manifolds,it suffices to solve the problem on reduced and simplified classes.
Lemma 3.6(Lemma 3.4,[33];[3])If A is reduced,then
As noted in Theorem 2.7,the symplectic genus is positive ifA·A>0.However,ifA·A<0,then it is possible for this quantity to be negative and thus to offer no apparent useful bound on the minimal genus.
Theorem 3.7(Theorem B,Theorem C,[33])LetA∈H2(M,Z)and assume thatAis primitive ifA·A=0.IfA·A≥η(A)−1,thenAis represented by a connected,embedded symplectic surface.
Observe that the inequality in the Lemma precisely ensures that the corresponding moduli spaces are non-empty.
From this,a characterization of certain classes representable by spheres can be obtained.
Theorem 3.8(Theorem 3,[31];Theorem 4.2,[33];Theorem 3,[17])Up toD(M)-equivalence,the following are the only classes with non-negative self-intersection which are spherically representable(k≥1):
For whenk≤9,the reduced case has been resolved completely.
Lemma 3.9(Theorem 1,[31])Assume thatAis reduced and 2≤k≤9.Then
B.H.Li [31]also produced an interesting result for multiple classes.
Lemma 3.10(Proposition 5,[31])If a reduced classAwithA·A≥0 hasgm(A)=η(A),then,for anyd>0,gm(dA)=η(dA).
For whenk≥10,there are still very few results.One result,obtained by B.H.Li and then used by Zhao-Gao-Qiu [52]to develop an explicit formula,is the following:
Lemma 3.11(Proposition 4,Proposition 6,[31])LetA0 be a reduced class withbi≤2 fori≥10.Thengm(A)=η(A)if either
(1)η(A)=1,or
(2)η(A)≥2,A·A≥0,andb10>0.
Note the restriction on the coefficientsbifori≥10.
In the simplified case,the key is the orbit structure from Lemma 3.5 and Theorem 2 in [32],as well as an explicit construction.
Theorem 3.12([4,34])LetM=andA∈H2(M,Z)withA2<0.Assume thatAis primitive ordinary and that theD(M)-orbitOAofAcontains a simplified class;then it has minimal genusgA=0.Moreover,for each class,there exists a complex orientation-compatible structure onMsuch that the surface can be chosen to be holomorphic with a symplectic orientation-compatible structure onMsuch that the surface is symplectic.
IfAis primitive characteristic,thing sareabit more complicated.In particular,the minimal genuscan only bedetermined for sufficiently many blow-ups;which is a stabilization-typeresult.
Theorem 3.13([4,34])LetM=andA∈H2(M,Z)withA2<0.Assume thatAis primitive characteristic,thatA2=−(8γ+k−1)(γ >0)and that theD(M)-orbitOAofAcontains a simplified class.
(1)If 1≤k≤2γ−1,then the minimal genus of any embedded surface representingAis bounded above by 2γ−k.
(2)Ifγis odd andk≥2γ−1,then the minimal genus of any embedded surface representingAis 1.
(3)Ifγis even andk≥2γ,then the minimal genus of any embedded surface representingAis 0.
The explicit classes in each category for 0>A·A>−17,3≤k≤9 were calculated in [33]and [34].This was extended to all simplified classes in [4].
Reduced classes withA·A<0 are not understood at the current time.
3.2 Irrational Ruled Manifolds
LetMbeirrational ruled.This means thatMis diffeomorphic to either a trivialS2-bundle over a surface Σgof genusg,the twistedS2bundle over Σg,denoted byN,or a blow-up of either of these base cases.The minimal genus problem for the first two cases is described in Theorems 20 and 21 of [26].Blow-ups of the two base cases are diffeomorphic.
Consider first the classA=bT.This class always has minimal genus 0 by tubing the b copies of the fiber.Moreover,theD(M)orbit of this class consists of two elements:bTand−bT.In what follows,this class will be excluded.The following definition is based on the results in [53]:
(1)The classAis called reduced if
(2)The classAis called simplified ifa=0 andbis the minimal non-negative value in theD(M)orbit containingA.
It is now possible to mimic the results for rational manifolds.In particular,analogous results hold for all statements up to Lemma 3.4.In particular,reduced and simplified orbits do not overlap,and simplified classes always have negative self-intersection,in a fashion analogous to the rational case.
Lemma 3.15([35,53])LetM=k≥1 andA∈H2(M,Z)withA/bT.
(1)AisD(M)-equivalent to a unique reduced or simplified class.
(2)Each orbitOAcontains either a simplified class or a reduced class,never both.There is an efficient algorithm to find a representative in each orbit.
Moreover,the following Theorem continues to hold:
Theorem 3.16(Theorem B,Theorem C,[33])LetA∈H2(M,Z)and assume thatAis primitive ifA·A=0.IfA·A≥η(A)−1,thenAis represented by a connected,embedded symplectic surface.IfA·A≥0,thenη(A)can be calculated usingKst.
Zhao-Gao [53]provided an explicit formula for the reduced class associated to anyA∈H2(M,Z).They then determined the minimal genus for certain classes using this reduction and an explicit construction while making use of the symplectic genus.
(1)Ais represented by an embedded sphere if and only ifa=0.
(2)Ifa0,then
(3)IfAis reduced,n≤4 andb≥0,thengm(A)=η(A).
B.H.Li and T.J.Li proved a similar result in the caseb<0:
The proof of this theorem makes use of a novel construction,called the circle sum.This operation takes two surfaces of genusg1andg2and produces a new surface of genusg1+g2−1.
LetΣ0and Σ1be closed,oriented surfaces of positive genusg0resp.g1disjointly embedded inM.By removing a disk from each and tubing them,one obtains the connected sum and a surface of genusg1+g2.Instead of considering embedded circles which bound a disk,consider embedded circleswhich represent a non-trivial class inH1(Σi,Z).Remove from each surface an annulusto obtain two surfaces with two boundary circles each.If the boundaries are pairwise connected by a tube,one either re-obtains the two disjoint surfaces or a new surface is formed.This is the circle sum of Σ1andΣ2,introduced in [36].
The key is to find an appropriate embedded annulus inMthat intersects the surfaces Σiat precisely the circlesand which admits a non-vanishing normal vector field that is tangential to Σibut normal toat the ends.Under these conditions,Theorem 2.1,[36],states that the circle sum exists,that the obtained surface has genusg1+g2−1,and that it is in the class [Σ0]+[Σ1].
3.3 Ellip tic Manifolds
The results in this section combine the techniques from above.Some version of an adjunction inequality is used to give a lower bound on the minimal genus,a Diff(M)-argument is used to reduce the set of classes that need to be considered,and finally,the circle sum is used to construct an explicit example,thus determining the minimal genus.
Hamilton [14]studied minimal,simply connected elliptic surfacesMwithb+>1.Letkdenote the Poincar´e dual of the canonical classKofMdivided by its divisibility(orkis the fiber class ifMis a K 3 surface).LetVdenote a class withk·V=1(for example,the class of a section of the fibration if there are no multiple fibers)withV·V=2bor 2b+1.Denote the classW=V−bk.Then the intersection form on the span ofWandkis either
ThenMhas intersection form given by either
withl≥2 andm>0.A(standard)basis for the secondHsummand can be obtained from the GompfnucleusN(2);see [11].This nucleus contains a rim torusRof self-intersection 0 and a sphere of self-intersection-2 such thatR·S=1.Choose as a basis of the secondHsummand the classesRandT=R+S,whereTcan be represented by a torus of self-intersection 0.
Building on work of Lönne [39],Hamilton obtained the following result:
Lemma 3.19(Proposition 2.10,[14])LetMbe an elliptic surface andBan arbitrary class in the subgrouplH⊕m(−E8)⊂H2(M,Z).ThenBcan be mapped to any other class inlH⊕m(−E8)of the same self-intersection and divisibility by a self-diffeomorphism ofM.In particular,Bcan be mapped to a linear combination ofRandS.This diffeomorphism acts by identity on the span〈k,W〉.
This diffeomorphism allows any class to be mapped to one that is contained inN(2),and thus the following lemma is crucial:
Lemma 3.20(Corollary 3.3,[14])Every non-zero class inH2(N(2),Z),not necessarily primitive,which has self-intersection 2c−2 withc≥0,is represented by an embedded surface of genuscinN(2).
This result follows from a result by Kronheimer-Mrowka;see Lemma 14,[26].Using the previous two results,the following is immediately apparent:
Lemma 3.21(Corollary 4.1,[14])LetMbe an elliptic surface.Then every non-zeroA∈H2(M,Z)of self-intersection 2c−2 withc≥0 that is orthogonal to the classesKandVis represented by a surface of minimal genusc.This surface can be assumed to be embedded inN(2)⊂M.
A slightly stronger version is possible ifMadmits no multiple fibers(in this casek=F,the class of a fiber).
Theorem 3.22(Theorem 1.1,Theorem 4.8,[14])LetMbe an elliptic surface without multiple fibers.Then every non-zeroA∈H2(M,Z)of self-intersection 2c−2 withc≥0 that is orthogonal toKis represented by a surface of minimal genusc.
The key to this proof is to reduce to the standard classαF+γ R+δTusing Lemma 3.19 and to use the circle sum construction to produce an embedded surface representing this class.
Nakashima [41]studiedT2-bundles over surfaces Σhof genush≥1.To begin,letM=T2×Σh.Denotex1,z1,···,xh,zhas a symplectic basis ofH1(Σh,Z)as described in Figure 1.Similarly,fi x a symplectic basisy,tofT2.LetSandFdenote the class of a section and of the fiber inH2(M,Z)respectively.WriteA∈H2(M,Z)as
Figure 1 The symplectic basis on Σh;see [41]
In this basis,the intersection form onMisH⊕2g+1.A detailed study of Diff(M)leads to the following result:
Lemma 3.23(Lemma 2.2,[41])For any homology classA∈H2(M,Z),there exists a self-diffeomorphism ofMsendingAto the class
for someα,β∈Z.
Note that the self-intersection is preserved,so the following must hold:
Using this reduction and the circle sum construction on cleverly constructed parallel tori leads to the following result:
Theorem 3.24(Theorem 1.2,[41])LetM=Σh×T2andA∈H2(M,Z),whereΣhis an oriented closed surface of genush≥1.Then
whereFis the fiber class and(*)means that one of the following conditions is satisfi ed:
This result can be re-stated in terms of the Thurston norm by viewingMas(Σh×S1)×S1(see Theorem(2.14)).
Corollary 3.25(Corollary 4.4,[41])LetM=(Σh×S1)×S1withh≥2 and letp:M→Σh×S1be the projection map.Then each classA∈H2(M,Z)satisfies
A similar splitting leads to the following result:
Theorem 3.26(Theorem 4.6,[41])Letp:N→Σhbe a non-trivialS1-bundle overΣhand letM=N×S1.Then,for each classA∈H2(M,Z),the minimal genus is given by
The next result,while not applicable to elliptic fibrations,fi ts nicely into the structure of Theorem 3.24.Note that the case considered by Edmonds [6],using only topological methods,corresponds to the last of the(*)cases withn=0.
Theorem 3.27(Theorem 1,[6])Let Σhand Σibe closed,orientable surfaces of a genus greater than 1.An elementA∈H2(Σh×Σi,Z)is represented by a topological or differentiable embedded torus Σ1if and only ifA=u⊗vfor someu∈H1(Σh,Z)andv∈H1(Σi,Z).
In Example 3.6,[37],two disjoint symplectic tori were constructed inM=Σ2×Σ2,representing the classesA1andA2,respectively.It was shown that no connected symplectic torusexistsin the classA1+A2,and Edmond’s result showsthat no smooth torus existseither.In particular,the circle sum construction cannot be applied to these two surfaces.
These results lead to the following speculation:
Sp eculation 3.28LetM=Σh×Σiand letA∈H2(M,Z).Then
The quantityb(A)is defined as follows(see also Definition 3.4,[37],for an equivalent description):recall the mapTfrom Eq.(2.1),
by
Thenb(A)is defined as
3.4 Disjoint Sp heres in Definite Manifolds with b+=2
In [25],Lawson prove the following Theorem:
Theorem 3.29(Theorem 17,[25])The only characteristic classes in CP2#CP2which are representable by spheres are(±1,±1).The only non-trivial divisible classes which can be represented by embedded spheres are
ford=2,3.The primitive and ordinary classes
can be represented by embedded spheres.
Conjecture 3.30(Conjecture1,[25])The only primitive and ordinary classes in CP2#CP2which can be represented by embedded spheres are given by Theorem 3.29.
Motivated by this Conjecture,letMbe a smooth,closed,connected,definite four-manifold withb1(M)=0 andb+(M)=2.ThenH2(M,Z)≃Z2and the classes in Theorem 3.29 still make sense.Applying Theorem 2.12,the following can be shown:
Lemma 3.31Assume thatA,B∈H2(M,Z)are non-trivial withA=(a1,a2).LetI={0,±1,±2}and assume that eithera1/∈Iora2/∈I.IfAandBare represented by disjoint connected surfaces,then at most one of the surfaces is an embedded sphere.
ProofDenote thatA=(a1,a2).As these are represented by disjoint surfaces ΣAand ΣB,A·B=0,and thusB=d(a2,−a1)for somed∈Q.
Assume thatAisa multiple class and that ΣA,ΣBare disjoint spheres.Using the results on multiple classes in Rokhlin [46],the only multiple classeswhich can be represented by embedded spheres are precisely the ones in Theorem 3.29.Assuming thatai/∈I,this leaves the classes
The pairings such thatA+Bcontains only odd entriescan not berepresented by disjoint spheres,as this would imply that either(3,1)or(3,3)are represented by embedded spheres.However,this is not the case on account of Kervaire-Milnor’s congruence and Rokhlin’s estimate.In the remaining cases,apply Theorem 2.12 to prove the claim;for example,suppose thatA=(3,0)andB=(0,2).Choosec=(3,1),and note thatc·c>σ(M),thatc·A>0 and thatc·B>0.Then
hence bothχ(Σa)andχ(ΣB)are non-positive.The remaining cases are similar.For example,ifA=(3,0)andB=(0,−2),choosec=(3,−1),and the same result follows.
Assume now thatBis a multiple class,but thatAis primitive.IfBcontains a 0,thenAcontains a±1 at the same spot;for example,ifB=(±3,0),thenA=(0,±1).Thus the classes with a 0 either lead to a classAwithai∈Ior aA+B=(3,1)-type class,which is not represented by an embedded sphere.This leavesB=(±2,±2).A calculation similar to that above again shows that both Euler characteristics in Theorem(2.12)are non-positive.For example,ifB=(2,−2),thenA=±(1,1).Choosec=(3,−1)or(1,−3);in both cases,
Assume finally that bothAandBare primitive.ThenA=(a1,a2)andB=±(a2,−a1).For the initial argument,choosec=(3,1)as the characteristic class and assume that both
are non-negative.Applying Theorem(2.12),either
must hold.Focus now on the right hand side terms:
Observe that on Z,x2±3x≥−2 andx2±x≥0.Moreover,switching between the signs reflects the curve about the y-axis.Thus,any estimates onxwill have their signs reversed under this switch.
This completes the initial argument where both equations in(3.2)arenon-negative.If both are non-positive,then use instead the class(−3,−1)to obtain
These are the same equations one would obtain forA=(−a1,−a2)in the argument using(3.5).Thus,the same result as before holds.
In order to handle the case in which the equations in(3.2)have opposite signs,use the classes(1,−3)and(−1,3).This is equivalent,in(3.3)and(3.4),to switchingAandBand one pair of signs.It thus does not change the result obtained above.
More precisely,ifB=±(a2,−a1)and
then usec=±(1,−3)to obtain
Furthermore,if
then usec=±(−1,3)to obtain
Consider now the functions arising forc=(1,−3):
These are identical to(3.5)for the classesA=(−a2,a1)andB=±(a1,a2).Hence the same estimates hold.
If we usec=(−1,3),then this is again the same as using the negative of the previous classes in the initial argument,thus applying the initial argument to the classes(a2,−a1)and−d(a1,a2). □
Lemma 3.32For any non-trivial classA=(a1,a2)∈H2(M,Z)not in the list of Theorem 3.29,Acannot be represented by two(or more)disjoint essential spheres.
ProofObserve that no three non-trivial classes can be orthogonal inM,asMis definite and hasb+=2.Assume thatAcan be represented by two disjoint spheres.WriteA=A1+A2withA1·A2=0 and assume thatAiis each represented by an embedded sphere.
(1)If one ofAisatisfies the conditions of Lemma 3.31,then any other surface,disjoint from the sphere representingAi,cannot be an embedded sphere.Thus it is not possible thatA1andA2are both represented by disjoint embedded spheres.
(2)This leaves that
Note that we may remove(0,0)and,as before,the condition on orthogonality implies that ifA1=(a,b),thenA2=d(b,−a)for somed∈Z.
Ifd=±1,then the only classes that can be obtained from this decomposition that are not in the list of Theorem 3.29 are(±1,±3),(±4,0)and(0,±4).The classes(±1,±3)are not represented by an embedded sphere due to Kervaire-Milnor’s congruence for characteristic classes.For the divisible classes(±4,0)and(0,±4),Rokhlin [46]bounded the genus below by 2.
Ifd=±2,then|a|,|b|≤1,or one of the components ofA2is too large and Lemma 3.31 applies.Then the only class that can be obtained from this decomposition that is not in the list of Theorem 3.29 is(±1,±3).As before,these classes are not represented by an embedded sphere. □
Corollary 3.33LetMbe a smooth,closed,connected,definite four-manifold withb1(M)=0 andb+(M)=2.IfAis not one of the classes in Theorem 3.29,then it admits no representation by two disjoint essential spheres.
These results imply a weaker version of Conjecture 3.30.Note also that this shows that the class(3,2)∈H2(CP2#CP2,Z)cannot be represented by two disjoint embedded spheres.It is still unknown if this class can be represented by an embedded sphere.
Nouh [42]studied certain classes of surfaces in CP2#CP2.The key argument here was to study surfaces in CP2B4with a boundary given by a knotK.These were then glued to a disk∆⊂CP2B4which has∂∆=K.Using this construction,upper bounds were determined for the genus of certain curves;for example ifn≥1,then
Similarly,ifn≥2,then
Furthermore,a minimal genus representative of(4,±1)was constructed using this technique.
In this section,a construction first described in [5]is presented.This turns out to be the circle sum in certain situations.Letπ:(E2n,ωE)→D2be a symplectic Lefschetz fibration.This means that
•(E,ωE)is a symplectic manifold with boundaryπ−1(∂D2);
•πhas finitely many critical pointsp0,···,pnaway from∂D2,whileπ−1(b)is a closed symplectic manifold symplectomorphic to(X,ω)whenbπ(pi)for anyi;
•fi x a complex structurejonD2.There is another complex structureJi,defined nearpi,so thatπis(Ji,j)-holomorphic in a holomorphic chart(z1,···,zn)nearpi,and under this chart,πhas a local expression
Take a regular value ofπ,b0∈D2as the base point.Suppose that one has a submanifoldZ2r−1⊂π−1(b0).ThenZhas isotropic dimension 1 if,at eachx∈Z,(TxZ)⊥ω∩TxZ=R〈vx〉.We callvxan isotropic vector atx;for example,relevant in what follows,a closed curve on a surface.
Suppose that we have a(based)Lefschetz fibration(E,π,b0)with a submanifoldZ⊂π−1(b0)of isotropic dimension 1.Letγ(t)⊂D2be a path withγ(0)=b0.Assume thatγ(t)/=π(pi)for alltandi.Notice that there is a natural symplectic connection onEin the complement of singular points as a distribution.Then,forxthe connection atxis defined by.HereTvEis the subbundle ofTEdefined by vertical tangent spacesT(π−1(π(x)))at pointx.E|γ=π−1(γ)thus inherits this connection and thus a trivialization by parallel transports.The symplectic connection also defines a unique lift ofγ′to a vector field ofE|γ.We will useπ−1(γ′)to represent this lift.
Now choose a vector fieldVonE|γtangential to the fibers;one obtains a flow defined byV+π−1(γ′).Suppose that the following holds:
Condition 4.1
•Zt⊂Eγ (t)is the timet-flow ofZ0=Z,and eachZtis of isotropic dimension 1.
Remark 4.2The tilted transport construction as described here can be easily generalized in many ways.A most interesting generalization is that one could admitVwith singularities,and thus change the topology ofZtwhentevolves.
4.1 A Local Variant of Tilted Transport and Symplectic Circle Sum
We explain next how to use a rather simple case of tilted transport to partly recover the circle sum construction in symplectic geometry.Note that the corresponding counterpart is well-known in the smooth category.
The setting under consideration is a pair of disjoint symplectic surfacesS0,S1⊂(M4,ω).Suppose that one has an open setU⊂Mso thatUS1×[−1,1]×D2(2),a trivial bundle overD2with annulus fibers,whileSi∩U=S1×[−1,1]×{i}.We claim that there is an embedded symplectic surfaceSwhich is the circle sum ofS0andS1.
As immediate consequence of the construction,by taking a finite number of nearby copies of generic fibers in an arbitrary Lefschetz fibration of dimension 4,is that one realizesn[F]as an embedded symplectic surface by performing symplectic circle sums on two consecutive copies.
AcknowledgementsWe thank Weiwei Wu for making us aware of Lambert-Cole’s results.