Modulo 2 counting of Klein-bottle leaves in smooth taut foliations 7 Boyu Zhang 1 0 2 n Abstract a J This article proves that the parity of the number of Klein-bottle 9 leaves in a smooth cooriented taut foliation is invariant under smooth ] deformations within taut foliations, provided that every Klein-bottle T leaf involved in the counting has non-trivial linear holonomy. G . h 1 Introduction t a m Itwasprovedin[1]thatforacoorientedfoliation,aC0-genericsmooth [ perturbationdestroys allclosed leaves withgenusgreater than1. This 1 v article explores theother sideof thestory. Itshows thatundercertain 4 conditions, one cannot get rid of Klein-bottle leaves of a taut foliation 8 by smooth deformations. 0 2 Let L be a smooth cooriented 2-dimensional foliation on a smooth 0 three manifold Y. The foliation L and the manifold Y are allowed to . 1 beunorientable. By definition, thefoliation Lis called ataut foliation 0 if for every point p ∈ Y there exists an embedded circle in Y passing 7 1 through p and being transverse to L. : v Definition 1.1. Let K ⊂ Y be a closed leaf of L. The leaf K is called i X nondegenerate if it has non-trivial linear holonomy. r a Consider a closed 2-dimensional submanifold K of Y. If K is cooriented, one can define an element PD[K]∈ Hom(H (Y;Z);Z) as 1 follows. Let [γ] be a homology class represented by a closed curve γ, then PD[K] maps [γ] to the oriented intersection number of γ and K. Since Hom(H (Y;Z);Z) ∼= H1(Y;Z), the element PD[K] 1 can be considered as an element of H1(Y;Z). If both Y and K are oriented and if the orientations of Y and K are compatible with the coorientation of K, the element PD[K] is equal to the Poincar´e dual of the fundamental class of K. 1 Definition 1.2. Let A∈ H1(Y;Z). A closed leaf K of L is said to be in the class A if PD[K] = A. The foliation L is called A-admissible if every Klein-bottle leaf of L in the class A is nondegenerate. The following result is the main theorem of this article. Theorem 1.3. Let A ∈ H1(Y;Z). Let L , s ∈ [0,1] be a smooth s family of coorientable taut foliations on Y. Suppose L and L are 0 1 both A-admissible. For i = 0,1, let n be the number of Klein-bottle i leaves in the class A. Then n and n have the same parity. 0 1 Notice that if there is no Klein-bottle leaf of L in the homology classA,thenLisautomaticallyA-admissible. Therefore,thefollowing result follows immediately. Corollary 1.4. Let A ∈ H1(Y;Z), and let L be an A-admissible smooth coorientable taut foliation on Y. Assume that L has an odd number of Klein-bottle leaves in the class A. Then every smooth de- formation of L through taut foliations has at least one Klein-bottle leaf in the class A. Itwouldbeinterestingtounderstandwhetherasimilarresultholds for torus leaves of taut foliations. For example, supposeL and L are 0 1 two oriented andcooriented taut foliations on Y that can bedeformed to each other through taut foliations. Suppose every closed torus leaf in a homology class e ∈ H (Y;Z) has non-trivial linear holonomy, is 2 it always true that the numbers of torus leaves in the homology class e in L and L have the same parity? The answer to this question is 0 1 not clear to the author at the time of writing this article. The article is organized as follows. Sections 2 and 3 build up nec- essary tools for the proof of theorem 1.3. Sections 4 and 5 prove the theorem. Section 6 gives an explicit example of corollary 1.4, con- structing a foliation with a Klein-bottle leaf that cannot be removed by deformations. I would like to express my most sincere gratitude to Cliff Taubes, who has been consistently providing me with inspiration, encourage- ment, and patient guidance. J 2 Moduli spaces of -holomorphic tori In [4], Taubes studied the behaviour of the moduli space of pseudo- holomorphic curves on a compact symplectic 4-manifold, and used it to define a version of Gromov invariant. This section recalls some results from [4] to prepare for the proof of theorem 1.3. The mod- uli space considered here is not exactly the moduli space used in the 2 definition of Taubes’s Gromov invariant, butessentially whatis devel- oped in this section is a special case of Taubes’s result. For a survey on different versions of Gromov invariants of symplectic 4-manifolds based on Taubes’s work, see [3]. Let X be a smooth 4-manifold. To avoid complications caused by exceptional spheres, assume throughout this section that π (X) = 0. 2 This will be enough for the proof of theorem 1.3. Let J be a smooth almost complex structure on X. Consider an immersed closed J-holomorphic curve C in X. Let N be the normal bundle of C, the fiber of N then inherits an almost complexstructurefromJ. Letπ :N → C betheprojectionfromN to C. Choose a local diffeomorphism ϕ from a neighborhood of the zero section ofN toaneighborhoodofC inX, whichmapsthezerosection of N to C. The map ϕ can be chosen in such a way that the tangent map is C-linear on the zero section of N. Every closed immersed J- holomorphic curve that is C1-close to C is the image of a section of N. Fix an arbitrary connection ∇ on N and let ∂¯ be the (0,1)-part 0 0 of ∇ . If s is a section of N near the zero section, the equation for 0 ϕ(s) to be a J-holomorphic curve in X can be schematically written as ∂¯ s+τ(s)(∇ (s))+Q(s)(∇ (s),∇ (s))+T(s)= 0. (2.1) 0 0 0 0 Here τ is a smooth section of π∗(HomR(T∗C⊗RN,T0,1C⊗CN)), and Q is a smooth section of π∗(HomR(T∗C ⊗R N ⊗R N,T0,1C ⊗C N)), and T is a smooth section of π∗(T0,1C⊗CN). The values of τ, Q, and T are defined pointwise by the values of J in an algebraic way, and τ, Q, T are zero when s = 0. The linearized equation of (2.1) at s = 0 is ∂¯ (s)+ ∂T(s)= 0. Define 0 ∂s ∂T L(s):= ∂¯ (s)+ (s). (2.2) 0 ∂s Notice that L is a real linear operator. The curve C is called nonde- generate if L is surjective as a map from L2(N) to L2(N). By elliptic 1 regularity, if C is nondegenerate then the operator L is surjective as a map from L2(N) to L2 (N) for every k ≥ 1. The index of the k k−1 operator L equals indL = hc (N),[C]i−hc (T0,1X),[C]i. (2.3) 1 1 It follows from the definition that nondegeneracy only depends on the 1-jet of J on C. Namely, if there is another almost complex struc- ture J′ such that (J −J′)| = 0 and (∇(J −J′))| = 0, then C is C C nondegenerate as a J-holomorpic curve if and only if it is nondegen- erate as a J′-holomorphic curve. 3 For a homology class e ∈ H (X;Z), define 2 d(e) = e·e−hc (T0,1X),ei. 1 By equation (2.3), d(e) is the formal dimension of the moduli space of embedded pseudo-holomorphic curves in X in the homology class e. By the adjunction formula, the genus g of such a curve satisfies e·e+2−2g = −hc (T0,1X),ei. 1 Therefore d(e) = 2(g − hc (T0,1X),ei − 1). In general, the formal 1 dimension of the moduli space of J-holomorphic maps from a genus g curve to X in the homology class e, modulo self-isomorphisms of the domain, is equal to 2(g−hc (T0,1X),ei−1). 1 Now assume X is has a symplectic structure ω. Recall that an almost complex structure J is compatible with ω if ω(·,J·) defines a Riemannianmetric. LetJ(X,ω)bethesetofsmoothalmostcomplex structures compatible with ω. For a closed surface Σ and a map ρ: Σ → X, define the topological energy of ρ to be ρ∗(ω). Σ R Definition 2.1. Let (X,ω) be a symplectic manifold. Let E > 0 be a constant. An almost complex structure J ∈ J(X,ω) is called E- admissible if the following conditions hold: 1. Every embedded J-holomorphic curve C with energy less than or equal to E and with d([C]) = 0 is nondegenerate. 2. For every homology class e ∈ H (X;Z), if h[ω],ei ≤ E, and if 2 hc (T0,1X),ei > 0 (namely, the formal dimension of the moduli 1 space of J-holomorphic maps from a torus to X in the homology class e, modulo self-isomorphisms of the domain, is negative), then there is no somewhere injective J-holomorphic map from a torus to X in the homology class e. The next lemma is a special case of proposition 7.1 in [5]. Recall that the C∞ topology on J(X,ω) is defined as the Fr´echet topology, namely it is induced by the distance function ∞ d(j ,j ) = 2−n · kj1 −j2kCn . 1 2 X 1+kj1−j2kCn n=1 Lemma 2.2. Let E > 0be a constant. If (X,ω) isa compact symplec- tic manifold, the set of E-admissible almost complex structures form an open and dense subset of J(X,ω) in the C∞-topology. A homology class e is called primitive if e 6= n·e′ for every integer n > 1 and every e′ ∈ H (X;Z). If e ∈ H (X;Z) is a primitive class, 2 2 4 define M(X,J,e) to be the set of embedded J-holomorphic tori in X with fundamental class e. Now consider smooth families of almost complex structures. As- sume ω (s ∈ [0,1]) is a smooth family of symplectic forms on X. For s i = 0,1, let J ∈ J(X,ω ). Define i i J(X,{ω },J ,J ) s 0 1 to be the set of smooth families {J } connecting J and J , such that s 0 1 J ∈ J(X,ω ) for each s ∈ [0,1]. The ideas of the following lemma s s can be found implicitly in [4]. Lemma 2.3. Let X be a compact 4-manifold and let ω (s ∈ [0,1]) s be a smooth family of symplectic forms on X. Let e ∈ H (X;Z) be a 2 primitive class with hc (T0,1X),ei = 0 and e·e = 0, and let E > 0 1 be a constant such that E > h[ω ],ei for every s. For i ∈ {0,1}, s let J ∈ J(X,ω ) be an E-admissible almost complex structure on X. i i Then there is an open and dense subset U ⊂ J(X,{ω },J ,J ) in the s 0 1 C∞-topology, such that for every element {J } ∈ U, the moduli space s M(X,{J },e) = M(X,J ,e) has the structure of a compact s s∈[0,1] s smooth 1-manifold`with boundary M(X,J ,e)∪M(X,J ,e). 0 1 Proof. The formal dimension of the moduli space of J -holomorphic s maps from a genus g curve to X in homology class e, modulo self- isomorphisms of the domain, is equal to 2(g − hc (T0,1X),ei − 1), 1 which always even. When the formal dimension is negative, it is less than or equal to −2. Therefore, there is an open and dense sub- set U ⊂ J(X,{ω },J ,J ), such that condition 2 of definition 2.1 ∞ s 0 1 holds for each J . The standard transversality argument shows that s on an open and dense subset U ⊂ U , the space M(X,{J },e) is 0 s a smooth 1-manifold with boundary M(X,J ,e)∪M(X,J ,e). For 0 1 general X and e the spcae M(X,{J },e) does not have to be com- s pact. However, since it is assumed that π (X) = 0, there is no non- 2 constant J -holomorphic maps from a sphere to X. By Gromov’s s compactness theorem (see forexample [6]), for every sequence {C } ⊂ n M(X,{J },e), there is a subsequence {C } with C ∈ M(X,J ,e) s ni ni si and lim s = s , such that the sequence C is convergent to one i→∞ i 0 ni of the following: (1) a branched multiple cover of a somewhere injec- tive J -holomorphic map, (2) a somewhere injective J -holomorphic s0 s0 map with bubbles or nodal singularities or both, (3) a somewhere injective J -holomorphic torus. Case (1) is impossible since e is as- s0 sumed to be a primitive class. Case (2) is impossible becase there is no non-constant J -holomorphic maps from a sphere to X. When s0 case (3) happens, for the limit curve the adjunction formula states that e · e + 2 − 2g = −hc (T0,1X),ei + κ, where κ depends on the 1 5 behaviour of singularities and self-intersections of the curve, and κ is always positive if the curve is not embedded (see [2]). Since g = 1, e ·e = 0, hc (T0,1X),ei = 0, it follows that κ = 0, hence the limit 1 curve is an embedded curve, namely it is an element of M(X,J ,e). s0 Therefore the space M(X,{J },e) is compact. s With a little more effort one can generalize lemma 2.3 to non- compact symplectic manifolds. To start, one needs the following defi- nition. Definition 2.4. Let (X,ω) be a symplectic manifold, not necessarily compact. Let J ∈ J(X,ω). The pair (ω,J) defines a Riemannian metric g on X. The triple (X,ω,J) is said to have bounded geometry with bounding constant N if the following conditions hold: 1. The metric g is complete. 2. The norm of the curvature tensor of g is less than N. 3. The injectivity radius of (X,g) is greater than 1/N. One says that a path {(X,ω ,J )} has uniformly bounded geome- s s try if each (X,ω ,J ) has bounded geometry, and the bounding con- s s stant N is independent of s. The following lemma is a well-known result. Lemma 2.5. Let (X,ω,J) be a triple with bounded geometry, with bounding constant N. Let e ∈ H (X;Z), and let E > 0 be a constant 2 such that E ≥ h[ω],ei. Then there is a constant M(N,E), depending only on N and E, such that every connected J-holomorphic curve C with fundamental class e has diameter less than M(N,E) with respect to the metric defined by ω(·,J·). Proof. By the monotonicity of area, there is a constant δ depending only on N, such that for every point p ∈ C the area of B (1/N)∩C is p greater thanδ. SinceC isconnected,thisimpliesthatthetotalareaof C boundsitsdiameter. NoticethattheareaofC equalsh[ω],ei, which is bounded by E, hence the the diameter is bounded by a function of N and E. In the noncompact case, one needs to be more careful about the topology of the spaces of almost complex structures. A topology on J(X,ω) can be defined as follows. Cover X by countably many com- pact sets {Ai}i∈Z. For each Ai define the C∞-topology on J(Ai,ω). Endow the product space J(A ,ω) i Yi∈Z 6 with the box topology, and consider the map J(X,ω) ֒−→ J(A ,ω) i Yi∈Z defined by restrictions. The topology on J(X,ω) is then defined as the pull back of the box topology on the product space. For N > 0, define J(X,ω,N) to be the set of almost complex structures J ∈ J(X,ω) such that (X,ω,J) has bounded geometry with bounding constant N. With the topology given above, the space J(X,ω,N) is an open subset of J(X,ω). A topology on J(X,{ω },J ,J ) can be defined in a similar way. s 0 1 Cover X by countably many compact sets {Ai}i∈Z. For each Ai define the C∞-topology on J(A ,{ω },J ,J ). The topology on the space i s 0 1 J(X,{ω },J ,J ) is then defined as the pull back of the box topology s 0 1 on the product space. For N > 0, define the set J(X,{ω },J ,J ,N) to be the set of s 0 1 families {J } ∈ J(X,{ω },J ,J ) such that {(X,J ,ω )} has uni- s s 0 1 s s formly bounded geometry with bounding constant N. Then the set J(X,{ω },J ,J ,N) is an open subset of J(X,{ω },J ,J ). s 0 1 s 0 1 The following lemma is essentially a diagonal argument. It ex- plains why the topologies defined above are the correct topologies to accommodate the perturbation arguments for the rest of the article. Lemma 2.6. Let {A } be a countable, locally finite cover of X n n≥1 by compact subsets. Let ω be a symplectic form on X, let ω be a s smooth family of symplectic forms on X. Let N > 0 be a constant. Let J ∈ J(X,ω ,N), where i = 0 or 1. i i 1. Let ϕ : J(X,ω) ֒−→ J(A ,ω) be the embedding map. For n n every n, let U be anQopen and dense subset of J(A ,ω), then n n ϕ−1( U ) is an open and dense subset of J(X,ω). n n Q 2. Let ϕ : J(X,{ω },J ,J ) ֒−→ J(A ,{ω },J ,J ) be the em- s 0 1 n i s 0 1 bedding map. For every n, letQ U ⊂ J(A ,{ω },J ,J ) n n s 0 1 be an open and dense subset, then ϕ−1( U ) is an open and n n dense subset of J(X,{ω },J ,J ). Q s 0 1 Proof. For part 1, the set ϕ−1( U ) is open by the definition of box n n topology. To prove that ϕ−1(Q U ) is dense, let J be an element n n of J(X,ω). Let J = J| ∈QJ(A ,ω). Let V ⊂ J(A ,ω) be n An n n n an arbitrary open neighborhood of J . One needs to find an element n J′ ∈ J(X,ω)suchthatJ′| ∈ V ∩U . Foreach n,letD beanopen An n n n neighborhood of A such that the family {D } is still a locally finite n n 7 cover of X. One obtains the desired J′ by perturbing J on the open sets {D } one by one. To start, perturbthe section J on D to obtain n 1 a section J . Since U is dense it is possible to find a perturbation 1 1 such that J | ∈ U ∩V . Now assume that after perturbation on 1 A1 1 1 D ,D ,··· ,D , one obtains a section J such that J | ∈ U ∩V for 1 2 k k k Ai j j j = 1,2,··· ,k. Then a perturbation of J on D gives a section k k+1 J such that J | ∈ U ∩V . When the perturbation is k+1 k+1 Ak+1 k+1 k+1 small enough, it still has the property that J | ∈ U ∩ V for k+1 Aj j j j = 1,2,··· ,k. Since {D } is a locally finite cover of X, on each n compact subset of X the sequence {J } stabilizes for sufficiently large k k. The limit lim J then gives the desired J′. k→∞ k Theproofsfor part2is exactly thesame, one only needsto change the notation J(·,ω) to J(·,{ω },J ,J ). s 0 1 Lemma 2.7. Let X be a 4-manifold, let e ∈ H (X;Z) be a primi- 2 tive class. Assume ω (s ∈ [0,1]) is a smooth family of symplectic s forms on X. Let E be a positive constant such that E > h[ω ],ei for s every s. For i = 0,1, assume J ∈ J(X,ω ,N) is E-admissible. If i i the set J(X,{ω },J ,J ,N) is not empty, then there is an open and s 0 1 dense subset U ⊂ J(X,{ω },J ,J ,N), such that for each {J } ∈ U, s 0 1 s the moduli space M(X,{J },e) = M(X,J ,e) has the struc- s s∈[0,1] s ture of a smooth 1-manifold with bo`undary M(X,J ,e)∪M(X,J ,e). 0 1 Moreover, if f : X → R is a smooth proper function on X, then the function defined as f : M(X,{J },e) → R s C 7→ fdA / 1dA Z Z (cid:0) C (cid:1) (cid:0) C (cid:1) is a smooth proper function on M(X,{J },e), where dA is the area s form of C. Proof. One first prove that there is an open and dense subset U ⊂ J(X,{ω },J ,J ,N), such that for every {J } ∈ U, the moduli space s 0 1 s M(X,{J },e) is asmooth 1-dimensional manifold. Letg bethe met- s s ric on X compatible with J and ω . Let g be a complete metric on X s s such that g ≥ g for every s. From now on, the distance function on s X is defined by the metric g. By lemma 2.5, there exists a constant M > 0 such that the diameter of every J -holomorphic curve with s energy no greater than E is bounded by M. Let {B } be a countable n locally finite cover of X by open balls of radius 1. For every n, let A n betheclosed ballwiththesamecenter as B andwithradius(M+1). n The family {A } is also a locally finite cover of X. For each n, let n M (X,{J },e) be the open subset of M(X,{J },e) consisting of the n s s curves C ∈ M(X,{J },e) such that C ∩ A 6= ∅. By the diameter s n 8 bound of J -holomorphic curves and the results for the compact case, s there is an open and dense subset U ⊂ J(A ,{ω },J ,J ,N) such n n s 0 1 thatif{J }| ∈U ,thenthesetM (X,{J },e)isasmooth1dimen- s An n n s sional manifold. It then follows from part 2 of lemma 2.6 that there is an open and dense subset U ⊂ J(X,{ω },J ,J ,N) such that for s 0 1 every element {J } ∈ U the set M(X,{J },e) is a smooth 1-manifold. s s When set M(X,{J },e) is a smooth 1-manifold, its boundary is s M(X,J ,e)∪M(X,J ,e), and the function f is a smooth function on 0 1 M(X,{J },e). s It remains to prove that f is a proper function. For any con- stant z > 0, take a sequence of curves C ∈ M(X,{J },e) such that n s |f(C )| < z. By the definition of f, there exists a sequence of points n p ∈ C such that |f(p )| < z. Since f is a proper function on X, n n n the sequence p is bounded on X. By lemma 2.5 this implies that n the curves C stay in a bounded subset of X. By the argument for n the compact case (lemma 2.3), the sequence {C } has a subsequence n that converges to another point in M(X,{J },e), hence function f is s proper. 3 Symplectization of taut foliations This section discusses a symplectization of oriented and cooriented taut foliations. It is the main ingredient for the proof of theorem 1.3. Let M be a smooth 3-manifold, let F be a smooth oriented and coorientedtautfoliationonM. SinceF iscooriented,itcanbewritten as F = kerλ where the positive normal direction of F is positive on λ. Since F is taut, there exists a closed 2-form ω such that ω∧λ > 0 everywhere on M. Choose a metric g on M such that ∗ λ = ω. 0 g0 By Frobenious theorem, dλ = µ∧λ for a unique 1-form µ satisfying µ ⊥ λ. Locally, write ω = e1 ∧e2 where e1 and e2 are orthonormal with respect to the metric g . Consider the 2-form Ω = ω+d(tλ) on 0 M ×R and the metric g defined by 1 g = ·(dt+tµ)2+(1+t2)λ2+(e1)2+(e2)2 1+t2 The 2-form Ω is a symplectic form on M × R, and the metric g is independent of the choice of {e1,e2} and is compatible with Ω. Let J be the almost complex structure given by (Ω,g). To simplify notations, let X be the manifold M ×R. Lemma 3.1 ([7], lemma 2.1). The triple (X,Ω,J) has bounded geom- etry. 9 Locally, let {e ,e ,e } be the basis of TM dual to {λ,e1,e2}, and 0 1 2 extend them to R-translation invariant vector fields on M ×R. Let eˆ = e −tµ(e )∂ , eˆ =e −tµ(e )∂ . The almost complex structure 1 1 1 ∂t 2 2 2 ∂t J is then given by ∂ 1 J = e , ∂t 1+t2 0 Jeˆ = eˆ . 1 2 Define F = span{eˆ ,eˆ }, it is a J-invariant plane field on X. 1 2 Lemma 3.e2. The plane field F is a foliation on X. Under the pro- jection M ×R→ M, the leaves of F projects to the leaves of F. e Proof. Since dµ∧λ = d(dλ) = 0, theere is a µ such that dµ = µ ∧λ. 1 1 Therefore, one has d(dt+tµ)= (dt+tµ)∧µ+tµ ∧λ, and dλ = µ∧λ. 1 By Frobenius theorem, the plane field F = ker(dt + tµ) ∩ kerλ is a foliation. The tangent planes of F projects isomorphically to the e tangent planes of F pointwise, thus the leaves of F projects to the e leaves of F. e Itturnsoutthat every closed J-holomorphiccurve inX is aclosed leaf of F. Lemmae 3.3. Let ρ : Σ → X be a J-holomorphic map from a closed Riemann surface to X. Then either ρ is a constant map, or it is a branched cover of a closed leaf of F. Proof. Since ρ is J-holomorphic, ρe∗ (dt+tµ)∧λ ≥ 0 pointwise on Σ. On the other hand, (cid:0) (cid:1) ρ∗ (dt+tµ)∧λ = ρ∗(d(tλ)) = 0. Z Z Σ (cid:0) (cid:1) Σ Therefore ρ(Σ) is tangent to ker(dt+tµ)∩kerλ, hence either ρ is a constant map, or it is a branched cover of a closed leaf of F. Lemma 3.4. Let L be a leaf of F and γ a closed curve oneL. Let π : M×R → M be the projection map. The foliation F is then transverse to π−1(γ) and gives a horizontal foliation on π−1(γ) ∼= γ ×R. The e holonomy of this foliation along γ is given by multiplication of l(γ)−1, where l(γ) is the linear holonomy of F along γ. Proof. Suppose γ is parametrized by u ∈ [0,1]. Let (γ(u),t(u)) be a curve in M×R that is a lift of γ and tangent to F. Then the function t(u) satisfies t˙+tµ(γ˙) = 0. Therefore e t(1) = e−R01µ(γ˙)dut(0). 10