Constructions of generalized complex structures in dimension four
CCONSTRUCTIONS OF GENERALIZED COMPLEXSTRUCTURES IN DIMENSION FOUR
RAFAEL TORRES
In this note, four-manifold theory is employed to study the existence of (twisted)generalized complex structures. It is shown that there exist (twisted) generalizedcomplex structures that have more than one type change loci. In an example-drivenfashion, (twisted) generalized complex structures are constructed on a myriad offour-manifolds, both simply and non-simply connected, which are neither complexnor symplectic. 1.
Introduction
Twisted generalized complex structures are a generalization of complex and sym-plectic structures introduced by Hitchin [20], and developed by Gualtieri in [17, 16].The existence of an almost-complex structure is the only known obstruction for theexistence of a twisted generalized complex structure on a manifold so far [16]. Giventhat surfaces are K¨ahler manifolds, the question of existence of such a structurebecomes non-trivial first in dimension four. In [9, 10], Cavalcanti and Gualtierishowed that generalized complex 4-manifolds form a larger set than symplecticand/or complex manifolds. They proved that a necessary and sufficient conditionfor the manifolds m CP n CP to admit a generalized complex structure, is thatthey admit an almost-complex one.The endeavor taken in this note is to study the existence of twisted generalizedcomplex structures using 4-manifold theory. A number of non-complex and non-symplectic twisted generalized complex manifolds are produced building on recentconstructions of small symplectic 4-manifolds [6, 12, 7, 3] by using the techniquesof [28, 24, 14, 4, 26, 9, 10] on Seiberg-Witten theory, symplectic and generalizedcomplex geometry. Among the results of the paper, there are the following. • A (twisted) generalized complex structure can have more than one typechange loci. • The connected sums m ( S × S ) , r ( S × S ) S × S , m CP n CP , r CP s CP S × S , L ( p, × S k CP admit a (twisted) generalized complex structure if and only if they have analmost complex structure. In particular, the generalized complex structures Date : November 24th, 2010.2010
Mathematics Subject Classification.
Primary 53C15, 53D18; Secondary 53D05, 57M50.
Key words and phrases.
Generalized complex geometry, type change loci, torus surgeries. a r X i v : . [ m a t h . DG ] A p r RAFAEL TORRES on m CP n CP built in this paper are different from the ones constructedin [10]. • Every finitely presented fundamental group is realized by a non-symplectictwisted generalized complex 4-manifold. • Constructions of twisted generalized complex 4-manifolds that do not ad-mit a symplectic nor a complex structure, and that have specific types offundamental groups. For example, abelian groups, free groups of arbitraryrank, and surface groups.The organization of the paper is the following. Section 2 contains a short in-troduction to twisted generalized complex structures. The main results used toequip four-manifolds with such a geometric structure, and a fundamental result onthe study of the Seiberg-Witten invariants are stated in Section 3. Generalizedcomplex structures for spin manifolds are constructed in Section 4. In Section 4.1,generalized complex structures on S × S with different numbers of type changeloci are constructed. The question of existence of a generalized complex structureon the connected sums (2 g − S × S ) is settled in Section 4.2, and a previewof existence results of twisted generalized complex structures on non-simply con-nected manifolds is given in Section 4.3. In Section 5, a large class of symplectic4-manifolds are put together in order to produce generalized complex structuresthat are neither complex nor symplectic. The sixth section is devoted to the studyof the existence of these structures within the non-simply connected realm. InSection 6.1, it is proven that all finitely presented groups are twisted generalizedcomplex, while being neither symplectic nor complex. The last part of the pa-per contains non-symplectic, non-complex examples of twisted generalized complexmanifolds with abelian, surface and free fundamental groups (Sections 6.2 and 6.3respectively). The paper ends with questions for further research in Section 7.2. Twisted generalized complex structures
Following the work of Gualtieri in [16], in this section we recall the basic defini-tions and examples of generalized complex structures.The
Courant bracket of sections of the bundle
T M ⊕ T ∗ M given by the directsum of the tangent and cotangent bundles of a smooth manifold M is[ X + ξ, Y + η ] H := [ X, Y ] + L X η − L Y ξ − d ( η ( X ) − ξ ( Y )) + i Y i X H ,where H is a closed 3-form on M .The bundle T M ⊕ T ∗ M can be be equipped with a natural symmetric pairingof signature ( n, n ) < X + ξ, Y + η > := ( η ( X ) + ξ ( Y ))as well. Definition 1. A twisted generalized complex structure ( M, H, J ) is a complexstructure J on the bundle T M ⊕ T ∗ M that satisfies the following two conditions.(1) It preserves the symmetric pairing.(2) Its + i -eigenbundle, L ⊂ T C M ⊕ T ∗ C M , is closed under the Courant bracket.The + i -eigenbundle L is a maximal isotropic subspace of T C M ⊕ T ∗ C M that satis-fies L ∩ L = { } . The bundle L can be used to fully describe the complex structure ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 3 J . Moreover, a maximal isotropic subspace L may be uniquely described by a linebundle K ⊂ ∧ • T ∗ C M . This complex line bundle K annihilated by the + i -eigenvalueof J is the canonical bundle of J .Indeed, the twisted generalized complex structure can be completely describedin terms of differential forms. In order for a complex differential form ρ ∈ ∧ • T ∗ C M to be a local generator of the canonical line bundle K of a twisted generalizedcomplex structure, and thus determine J uniquely, it is required that ρ satisfiesthe following three properties at every point p ∈ M . • Algebraic property: the form can be written as ρ = e B + iω ∧ Ω,where
B, ω are real 2-forms, and Ω is a decomposable complex form. • Non-degeneracy: the non-vanishing condition( ρ, ρ ) = Ω ∧ Ω ∧ (2 iω ) n − k (cid:54) = 0holds. Here 2 n = dim ( M ), and k = deg (Ω). • Integrability: the form ρ is integrable in the sense that dρ + H ∧ ρ = ( X + ξ ) · ρ ,for a section X + ξ of T M ⊕ T ∗ M .The non-degeneracy requirement is equivalent to the condition L ∩ L = { } .It implies that at each point of a twisted generalized complex manifold, the realsubspace ker Ω ∧ ω ⊂ T M inherits a symplectic structure from the 2-form ω , and atransverse complex structure is defined by the annihilator of ω , as the + i -eigenspaceof a complex structure on T /ker Ω ∧ Ω. Definition 2.
Type and parity of a twisted generalized complex structure . Let ρ = e B + iω ∧ Ω be a generator of the canonical bundle K of a generalized complexstructure J at a point p ∈ M . The type of J at p is the degree of Ω. The parity of J is the parity of its type.The type of a twisted generalized complex structure need not be constant, itmay jump along a codimension two submanifold. However, the parity of the typedoes remain the same along connected components of the manifold M . Remark 3. (Twisted) generalized complex structures: on the 3-form H of ourconstructions. For the twisted generalized complex structures (
M, H, J ) producedin this paper, the 3-form H is always given by a generator of H ( M ; Z ). Poincar´eduality implies that if a 4-manifold is simply connected, then the 3-form satisfies H = 0. In this case, a generalized complex structure is obtained. In particular,the non-simply connected manifolds constructed will have a twisted generalizedcomplex structure. For example, if the fundamental group is π ( ˆ M ) ∼ = Z , then the3-form H is a generator of H ( M ; Z ) ∼ = Z . Two standard examples of generalized complex manifolds are the following.
Example 4.
Complex and symplectic manifolds . Let ( M n , I ) be a complex mani-fold. Then M n is a generalized complex manifold of type n . Indeed, we can defineon T M ⊕ T ∗ M , the operator RAFAEL TORRES J I := (cid:18) − I I ∗ (cid:19) . In this case, T , M ⊕ T ∗ , M is the + i -eigenspace of J I , and the canonical bun-dle is K = ∧ n, T ∗ M .A symplectic manifold ( M, ω ) is a generalized complex manifold of type 0. Theoperator J ω := (cid:18) − ω − ω (cid:19) is a complex structure on the bundle T M ⊕ T ∗ M , whose + i -eigenspace is given by { X − iω ( X ) : X ∈ T C M } ; the canonical bundle K is generated by the form e iω .Regarding the jump on the type of a twisted generalized complex structure, thefollowing is observed. The projection ∧ • T ∗ C M → ∧ T ∗ C M determines a canonical section s of K ∗ . In four dimensions, the vanishing of thissection forces the type of a twisted generalized complex structure to jump from 0to 2. The jump occurs along a 2-torus, which inherits a complex structure. Definition 5.
A point p in the type-changing locus of a twisted generalized complexstructure on a 4-manifold is a nondegenerate point if it is a nondegenerate zero of s ∈ C ∞ ( K ∗ ).Regarding the notion of submanifolds in the generalized complex setting, wehave the following. Definition 6.
Branes . Let (
M, H, J ) be a twisted generalized complex manifold,and let i : Σ (cid:44) → M be a submanifold with a 2-form F ∈ Ω (Σ) that satisfies dF = i ∗ H , and such that the sub line bundle τ F ⊂ ( T M ⊕ T ∗ M ) | N defined as τ F := { X + ξ ∈ T Σ ⊕ T ∗ M : i ∗ ξ = i X F } ,is invariant under J , i.e., τ F is a complex sub-bundle. Such a submanifold Σ iscalled a brane .In the case where M is a complex manifold, the definition of a brane coincideswith the notion of a complex submanifold. Analogously, for symplectic manifolds,Lagrangian submanifolds provide examples of branes.3. Tools and techniques of construction
The surgical methods used in the production of twisted generalized complexstructures on 4-manifolds are introduced in this section.3.1.
Torus surgeries.
Let T be a 2-torus of self intersection zero inside a 4-manifold X . There is a diffeomorphism N T → T × D from its tubular neigh-borhood N T to a thick 2-torus T × D . Let { α, β } be the generators of π ( T ) andconsider the meridian µ T of T inside X − T , and the push offs S α , S β in ∂N T ≈ T .There is no ambiguity regarding the choice of push offs, since in our constructionsthe manifold X will be symplectic, the torus T will be Lagrangian, and the pushoffs are taken with respect to the Lagrangian framing. The loops S α and S β are ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 5 homologous in N T to α and β respectively. In particular, the set { S α , S β , µ } formsa basis for H ( ∂N ; Z ) ∼ = H ( T ; Z ).The manifold obtained from X by performing a ( p, q, r ) - torus surgery on T along the curve γ := pS α qS β is defined as X T,γ ( p, q, r ) = ( X − N T ) ∪ φ ( T × D ),where the diffeomorphism φ : T × ∂D → ∂ ( X − N T ) used to glue the piecestogether satisfies φ ∗ ([ ∂D ]) = p [ S α ] + q [ S β ] + r [ µ T ] in H ( ∂ ( X − N T )); Z ).A few words for the reader to get comfortable with our notation are in order.A (0 , ,
1) -torus surgery on T amounts to carving the 2-torus out, and gluing itback in exactly the same way. Therefore, if one performs a (0 , ,
1) - torus surgeryalong a torus T on X , one has X T,γ (0 , ,
1) = X . Whenever p = 0 = q , the surgerycoefficients ( p, q, r ) = (0 , ,
1) are said to be trivial.The Euler characteristic and signature of a 4-manifold are invariant under torussurgeries. If the torus T is essential and the surgery coefficients are nontrivial, then b ( X T,γ ( p, q, r )) = b ( X ) − b ( X T,γ ( p, q, r )) = b ( X ) − S α := m and S β := l . Lemma 7.
The fundamental group of the manifold obtained by applying a ( p, q, r ) - torus surgery to X on the torus T along the curve m + l is given by the quotient π ( X T,γ ( p, q, r )) ∼ = π ( X − T ) /N ( µ rT m pT l qT ) ,where N ( µ rT m pT l qT ) denotes the normal subgroup generated by µ rT m pT l qT .Proof. cf. [6, Proof Lemma 4]. We argue in terms of the effect that the attachmentof n -handles has on the fundamental group. The manifold T × D has a handlebodydecomposition consisting of one 0-handle, two 1-handles, and one 2-handle. Usingthe dual decomposition, in order to glue T × D back in, one attaches one 2-handle,two 3-handles, and one 4-handle. The fundamental group of the resulting manifoldchanges as in the statement of the lemma by attaching the 2-handle. Attaching 3-and 4-handles has no effect on the fundamental group of the resulting manifold. (cid:3) Provided that certain hypothesis on the manifold that undergoes surgery andon the corresponding torus hold, a geometric structure is readily available for X T,γ ( p, q, r ). If X admits a symplectic form for which the torus T is Lagrangian,then performing a ( p, q, N T results in X T,γ ( p, q,
1) being symplectic [4]; this surgical procedure is known as
Luttinger surgery [25, 4]. The next section is devoted to describe the existence oftwisted generalized complex structures on X T,γ ( p, q, p, q, r ) already encode the surgery curve γ , from nowon it will be dropped from our notation: X T ( p, q, r ) := X T,γ ( p, q, r ). RAFAEL TORRES
Surgical procedures to endow manifolds with a generalized complexstructure.
In [9] and [10], Cavalcanti and Gualtieri have studied and employedclassic topological procedures in order to equip a 4-manifold with a twisted gener-alized complex structure.Their main results that will be used for our purposes are recalled in the followingthree theorems.
Theorem 8. [9, Theorem 3.1 and Corollary 1] , [10, Theorem 4.1] (p, q, 0)-torussurgery . Let ( M, ω ) be a symplectic 4-manifold, which contains a symplectic torus T of self-intersection zero and of area A . Let ˆ M := M T ( p, q, be the result ofperforming a ( p, q, -torus surgery to M along T . Then ˆ M admits a twisted gen-eralized complex structure such that • The complex locus is given by an elliptic curve Σ with modular parameter τ = i , and the induced holomorphic differential Ω has periods < A, iA > . • Integrability holds with respect to a 3-form H , which is Poincar´e dual to A times the homology class of an integral circle of Re (Ω − ) in Σ .The ( p, q, - torus surgery can be performed simultaneously on a collection ofdisjoint symplectic tori in M . Each time such a surgery is performed, a type change locus is obtained in thetwisted generalized complex 4-manifold M T ( p, q, m CP n CP are obtainedby applying one ( p, q,
0) surgery to an elliptic surface E ( m −
1) along a torus fiber.Much like in the symplectic and complex scenarios, the existence of a general-ized complex structure on a manifold is preserved under the blow up/blow downoperations. The changes on the ambient manifold are exactly the same as in thecomplex/symplectic case.
Theorem 9. [9, Theorem 3.3] Blow ups . For any non-degenerate complex point p ∈ M in a twisted generalized complex 4-manifold M , there exists a twisted general-ized complex 4-manifold ˜ M and a generalized complex holomorphic map π : ˜ M → M that is an isomorphism ˜ M − π − ( { p } ) → M − { p } . The pair ( ˜ M , π ) is called theblow up of M at p , and it is unique up to canonical isomorphism. Theorem 10. [9, Theorem 3.4] Blow downs . A twisted generalized complex4-manifold ˜ M containing a 2-brane Σ = S intersecting the complex locus in a singlenon-degenerate point may be blown down to a generalized complex 4-manifold M .That is, there is a generalized holomorphic map π : ˜ M → M to a twisted generalizedcomplex manifold M that is an isomorphism ˜ M − Σ → M − { p = π (Σ) } . On the Seiberg-Witten invariants of the manifolds manufactured.
A purpose of this paper is to enlarge the class of twisted generalized complex 4-manifolds that are neither symplectic nor complex. We wish to make sure that themanifolds constructed have trivial Seiberg-Witten invariant [29]. From the work ofTaubes [28], such manifolds will not admit a symplectic structure.The basic result for such a purpose is the adjunction inequality.
ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 7
Theorem 11.
Adjunction inequality [24, 26] . Let X be a smooth closed oriented4-manifold with b +2 ( X ) > , and let Σ g (cid:44) → X be a smoothly embedded genus g surface with non-negative self-intersection. If g ≥ , then every basic class K of X satisfies g − ≥ | < K, [Σ g ] > | + [Σ g ] · [Σ g ] .Furthermore, if g = 0 and [Σ g ] ∈ H ( X ; Z ) is not a torsion class, then theSeiberg-Witten invariant of X vanishes. Generalized complex structures on spin manifolds
In this section, we occupy ourselves with equipping the almost-complex con-nected sums of copies of S × S with generalized complex structures. The general-ized complex structures built on S × S are new, and we observe some unexpectedphenomena concerning the number of type changing loci. Our constructions oweplenty to the efforts of Akhmedov, Baldridge, Fintushel, Kirk, D. Park, and Sternin [7, 12, 3] to construct exotic simply connected 4-manifolds with small Eulercharacteristics.4.1. More generalized complex structures on S × S . Perturbing the K¨ahlerstructure on CP × CP yields a generalized complex structure on S × S , whichhas a single type change locus. In this section we to construct different generalizedcomplex structures on S × S , in the sense that they have more than one typechange loci. We are indebted to Gil Cavalcanti and to Marco Gualtieri for pointingthis out.Consider the product of two copies of a genus 2 surface endowed with the productsymplectic structure Σ × Σ . Denote the loops generating the fundamental group a i × { s } , b i × { s } , { s } × c i , { s } × d i by a , b , a , b and c , d , c , d , so that π (Σ ) = < a , b , a , b | [ a , b ][ a , b ] = 1 > for the first factor, and analogously π (Σ ) = < c , d , c , d | [ c , d ][ c , d ] = 1 > for the second factor. The topologicalproperties that will be used are summarized in the following proposition. Proposition 12. (cf. [12, Section 4]) . The first homology group is given by H (Σ × Σ ; Z ) ∼ = Z , and it is generated by the loops a , b , a , b , c , d , c , and d . The second homology group H (Σ × Σ ; Z ) = Z is generated by the sixteentori a i × c j , a i × d j , b i × c j , b i × d j ( i = 1 , , j = 1 , , and the surfaces Σ × { pt } , { pt } × Σ . The tori a i × c j , b i × c j and the surface Σ × { pt } are geometrically dualto the tori b i × d j , a i × d j and the surface { pt } × Σ respectively, in the sense that a i × c j and b i × d j intersect transversally at one point (and similarly for the othersurfaces), and every other intersection is pairwise empty. The intersection formover the integers is even, and it is generated by nine hyperbolic summands.The Euler characteristic and signature of the product manifold are given by e (Σ × Σ ) = 4 , and σ (Σ × Σ ) = 0 . Notice that the element in the first homology group and the loop generating itare being denoted by the same symbol. More homotopical information on theseloops will be needed in our constructions, and we will proceed as follows. Remov-ing a surface from a 4-manifold may change the fundamental group of the ambient
RAFAEL TORRES manifold, since non nullhomotopic loops might be introduced by the meridian ofthe surface, thus adding a generator to the presentation of the group.According to Fintushel, D. Park and Stern [12, Section 4] inside the complementin Σ × Σ of eight of the Lagrangian tori described above, one can choose basepathsfrom a basepoint ( x, y ) of Σ × Σ to a basepoint of the boundaries of the tubularneighborhoods of the tori so that the two Lagrangian push offs, and the meridiansof the tori are given by { ˜ a , ˜ c ; µ = [˜ b − , ˜ d − ] } , { ˜ a , ˜ c ; µ = [˜ b − , ˜ d − ] } , { ˜ a , ˜ c ; µ = [˜ b − , ˜ d − ] } , { ˜ a , ˜ c ; µ = [˜ b − , ˜ d − ] } , { ˜ b , ˜ d ˜ c ˜ d − ; µ = [˜ a − , ˜ d ] } , { ˜ b , ˜ d ˜ c ˜ d − ; µ = [˜ a − , ˜ d ] } , { ˜ b ˜ a ˜ b − , ˜ d ; µ = [˜ b , ˜ c − ] } , { ˜ b ˜ a ˜ b − , ˜ d ; µ = [˜ b , ˜ c − ] } .The elements with tildes denote loops that are homotopic to the correspondingloop that generates an element in homology. In [12, Section 4], homotopies amongthe loops are found in order to obtain a presentation for the fundamental group ofcomplement of the tori inside Σ × Σ that is still generated by the same numberof generators that π (Σ × Σ ) has. Remark 13.
Abuse of notation.
From now on we will make no distinction be-tween a loop, its homotopy class, and the corresponding generator in homology.In particular the decorations above will be abandoned: a loop ˜ a will now just bedenoted by a . These oversimplifications are justified by following and building uponrecent papers [6, 12, 7, 3], where torus surgeries are used to unveil exotic smoothstructures on almost-complex 4-manifolds. In particular, we are helped greatly bythe analysis done by Baldridge and Kirk in [6, 7] . Proposition 12 says that the torus, its meridian and its Lagrangian pushoffsavailable for surgery are given by T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := b × c , m = b , l = d c d − , µ = [ a − , d ], T := b × c , m = b , l = d c d − µ = [ a − , d ], T := a × d , m = b a b − , l = d , µ = [ b , c − ], and T := a × d , m = b a b − , l = d , µ = [ b , c − ].The reader is kindly reminded that in our notation, for example, a ( p, , r )-torussurgery on T stands for a torus surgery on T along the curve m p = a p ; if r = 1,this torus surgery is a Luttinger surgery [4], and the case r = 0 correspond to thesurgery described in Theorem 8.. We construct generalized complex structures on S × S with a prescribed number of type change loci (up to eight of them) asfollows. We exemplify two cases.Say we would like a generalized complex structure with a number of eighttype change loci. Use Gompf’s result [14, Lemma 1.6] to perturb the symplec-tic form on Σ × Σ , so that the eight homologically essential Lagrangian tori { T , . . . , T } become symplectic. Perform simultaneously (1 , , T , T , T , T , and (0 , , T , T , T , T . Let X (8) be the ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 9 manifold obtained after the surgeries, which is diffeomorphic to S × S . By Theo-rem 8, X (8) admits a generalized complex structure that contains eight type changeloci.Let us now produce a generalized complex structure on S × S that containsfour type change loci. Our starting symplectic manifold is again Σ × Σ . Perturbthe symplectic form so that the homologically essential Lagrangian tori T , T , T and T become symplectic tori [14, Lemma 1.6]. Perform four Luttinger surg-eries: (0 , , +1)-surgery on T , (1 , , − T , (1 , , +1)-surgery on T ,(0 , , − T , and (1 , , +1)-surgeries on T and on T . Now performfour ( p, q, , , T and on T , and (0 , , T andon T . Denote the resulting manifold by X (4).By Theorem 8, X (4) admits a generalized complex structure. The group π ( X )is generated by the elements a , b , a , b , c , d , c , d , and the following (amongothers) relations hold has the following presentation: c = [ d − , b − ] , a = [ b − , d − ] , a = [ d − , b − ] , c = [ b − , d − ], b = [ d , a − ] = 1 , b = [ d , a − ] = 1 , d = [ c − , b ] = 1 , d = [ c − , b ] = 1.It is straightforward to see that π ( X (4)) = { } . Torus surgeries preserve boththe Euler characteristic and the signature. Thus, e ( X (4)) = 4 and σ ( X (4)) = 0.Moreover, the generalized complex manifold X is spin. Indeed, the homological ef-fect of such a ( p, ,
0) or (0 , q, X (4) is homeomorphic to S × S . The four type change loci arise as the core toriof the last four torus surgeries.Inside the symplectic 4-manifold obtained by applying the four Luttinger surg-eries are the symplectically imbedded surfaces of genus two Σ × { x } , and { x } × Σ .The ( p, , , q, X (4) is diffeomorphicto S × S .Variations on the surgical procedure described above yield the following theorem Theorem 14.
Let n ∈ { , , , , , , , } . The 4-manifold S × S admits ageneralized complex structure ( X ( n ) , H, J ) , which has n type change loci. The almost-complex connected sums (2 g − S × S ) , g ≥ . We gener-alize the procedure described in the previous section with the purpose of endowingthe connect sums of S × S that are almost-complex with a generalized complexstructure. The first step is to consider the product Σ × Σ g of a genus 2 surface witha genus g ≥ a i , b i , c j and d j ( i = 1 , j = 1 , . . . , g ) be the standard generators of π (Σ ) and π (Σ g ) respec-tively. We encode the topological information we need in the following proposition.In the notation regarding the fundamental group of the manifolds that undergo surgery, each relation is associated to the torus on which a surgery introduces theaforementioned relation. Proposition 15. (cf. [3, Section 2]) . The first homology group is given by H (Σ × Σ g ; Z ) = Z g , and it is generated by the loops a , b , a , b , c , d , · · · c g , d g .The second homology group H (Σ × Σ g ; Z ) = Z g +2 is generated by g tori, andthe surfaces [Σ × { pt } ] , [ { pt } × Σ g ] : the tori a i × c j and a i × d j are geometricallydual to b i × d j and b i × c j respectively, and the surface Σ × { pt } is geometricallydual to { pt } × Σ g . The intersection form over the integers is even, and it is givenby g + 1 hyperbolic summands; the pairs of tori contribute g summands, and onesummand is contributed by the surfaces Σ ×{ pt } and { pt }× Σ g . The characteristicnumbers are given by e (Σ × Σ g ) = 4 g − , and σ (Σ × Σ g ) = 0 .The manifold X g with b ( X g ) = 0 obtained from applying g ( p, q, r ) -torussurgeries to Σ × Σ g ( p, q ∈ { , } , p (cid:54) = q ) has fundamental group generated by theelements a , b , a , b , c , d , . . . , c n , d n , and the following relations hold. T : a = [ b − , d − ] r , T : b = [ a − , d ] r , T : a = [ b − , d − ] r , T : b = [ a − , d ] r T : c = [ d − , b − ] r , T : d = [ c − , b ] r , T : c = [ d − , b − ] r , T : d = [ c − , b ] r T : c = [ a − , d − ] r , T : d = [ a − , c − ] r , . . . , T g : c g = [ a − , d − g ] r , T g : d g = [ a − , c − g ] r . The proof is a generalization of [12, Section 4]. The claims concerning the sub-manifolds, and their homological properties are left to the reader. A proof forthe nontrivial claim regarding the fundamental group calculations can be found in[21]. Notice that in the notation of Proposition 15, the relation introduced to thefundamental group by the corresponding ( p, q, r )-surgery appears associated to itscorresponding homologically essential torus. For example, T : a = [ b − , d − ] stands for the relation coming from a (1 , , , , T is then T : a = [ b − , d − ] = 1.We proceed to construct a generalized complex structure on the connected sums(2 g − S × S ) for g ≥
3. Using [14, Lemma 1.6], we can perturb the symplec-tic form on Σ × Σ g so that the tori { T , T , T , · · · T g } become symplectic [14,Lemma 1.6]. Let X g be the manifold of Proposition 15 obtained by applying torussurgeries with r = 0. Theorem 8 implies that X g admits a generalized complexstructure.By the presentation of the group π ( X g ) given in Proposition 15, this impliesthat the manifold X g is simply connected. A direct computation of the char-acteristic numbers yields e ( X g ) = e (Σ × Σ g ) = − · (2 − g ) = 4 g −
4, and σ ( X g ) = σ (Σ × Σ g ) = 0. By Freedman’s Theorem [13], X g has the homeomor-phism type of (2 g − S × S ).We are ready to prove the following result. Theorem 16.
Let m ≥ . The connected sum m ( S × S ) admits a generalized complex structure if and only if it admits an almost-complexstructure. ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 11
In particular, m = (2 g −
3) is an odd number. The necessity of the existenceof an almost-complex structure is clear [16]. Concerning the sufficiency of thecondition, it remains to be demonstrated that the manifold X g constructed aboveis (2 g − S × S ). Proof.
Theorem 8 implies X g admits a generalized complex structure. We claimthat X g is diffeomorphic to (2 g − S × S ). For the sake of clarity, we work out thecase g = 3 by looking at the effect the surgeries have on the submanifolds of Σ × Σ ;the proofs for the cases g > × Σ transform the imbedded surfaces Σ ×{ x } and { x }× Σ into spheres, by reducing their genus. Moreover, according to Proposition 15,there are four remaining homologically essential submanifolds that are geometricallydual. Inside Σ × Σ , these are tori, which after the surgeries become geometricallydual 2-spheres. For example, the (1 , , T reduces the genus of the tori a × c , a × c , and the genus of their dual tori as well. Thus, becoming imbedded2-spheres. Each of these pairs span an S × S summand in X g . Therefore, X isdiffeomorphic to 3( S × S ). The Seiberg-Witten invariant of X is trivial, and it isnot symplectic. It is not a complex manifold by Kodaira’s classification of complexsurfaces [22, 8]. (cid:3) Remark 17.
Number of loci.
Let g ≥
3. The generalized complex structures builtin Theorem 16 have a number of 4 + 2 g type change loci. Following the methoddescribed in Section 4.1, generalized complex structures with a different number ofloci can be produced.4.3. Preview of other fundamental groups.
Constructions of non-simply con-nected 4-manifolds are given in Section 6. However, At this stage we would like topoint out that a variation on the coefficients of the torus surgeries in the previousconstructions promptly yields twisted generalized complex 4-manifolds with severalother fundamental groups. A sample for abelian groups is the following.
Proposition 18.
Let g ≥ . The homeomorphism types corresponding to • π = Z /p Z : (2 g − S × S ) (cid:94) L ( p, × S • π = Z /p Z ⊕ Z /q Z : (2 g − S × S ) (cid:92) L ( p, × S • π = Z : (2 g )( S × S ) S × S admit a twisted generalized complex structure, which does not come from a sym-plectic nor a complex structure. The pieces (cid:94) L ( p, × S and (cid:92) L ( p, × S stand for the manifolds obtained bymodifying the product L ( p, × S of a Lens space with a circle as follows. Performa surgery on L ( p, × S along { x } × α ( x ∈ L ( p, π = Z /p Z of the resultingmanifold comes from the fundamental group of the Lens space. This amounts toremoving a neighborhood of the loop S × D and glueing in a S × D . Denotesuch a manifold by (cid:94) L ( p, × S . Applying the same procedure to the loop { x } × α q results in a 4-manifold with π = Z /p Z ⊕ Z /q Z . Such manifold is denoted by (cid:92) L ( p, × S . Proof.
These manifolds are constructed by changing the surgery coefficients on oneor two of the torus surgeries used in the proof of Theorem 16 following Lemma 7.Generalized complex manifolds with infinite cyclic fundamental group are built by not performing one of the torus surgeries. Twisted generalized complex manifoldswith finite cyclic fundamental group are built by changing one of the surgery coeffi-cients (1 , ,
0) for a ( p, ,
0) with p (cid:54) = 0. The homeomorphism criteria in the infinitecyclic fundamental group case is due to Hambleton and Teichner [19, Corollary 3].If p >
1, then one obtains a twisted generalized complex manifold with finite cyclicfundamental groups, and its universal cover is nonspin. The 3-form H is given bythe generator of H . The corresponding homeomorphism criteria is due to Ham-bleton and Kreck, and it is given in [18, Theorem C]. The case π = Z /p Z ⊕ Z /q Z isleft as an exercise; the homeomorphism criteria, also due to Hambleton and Kreck,is given in [18, Theorem B].The argument minding the lack of a symplectic or a complex structure is verba-tim to the one given for Theorem 16. (cid:3) Constructions of non-symplectic and non-complex generalizedcomplex manifolds through the assemblage of symplectic ones
As an immediate corollary to the main result of the previous section, we constructnew generalized complex structures on non-spin simply connected almost-complex4-manifolds. We also recover an existence result of Cavalcanti and Gualtieri.
Proposition 19. (cf. [10, Section 5]) . Let n ≥ m . The manifolds m CP n CP admit a generalized complex structure if and only if they admit an almost-complexone. Moreover, there exist generalized complex structures on m CP n CP withmore than one type change loci. In particular, m is an odd number. We set m = 2 g −
3. Blowing up (Theorem 9)and/or blowing down (Theorem 10) (2 g − S × S ) produces generalized complexstructures on (2 g − CP n CP that can be chosen to have as many as 4 + 2 g typechange loci. As it was mentioned in Remark 17, the construction can be modifiedto obtain a different number of loci (cf. Section 4.1). For the sake of simplicity,we will give the argument for the case g = 3; the other cases follow verbatim. Thesymbol X = Y indicates a diffeomorphism between manifolds X and Y . Proof.
The necessity of the existence of an almost complex structure is clear [16].Let X be the blow up of the generalized complex manifold 3( S × S ). Theorem9 implies X is generalized complex. We have X = 3( S × S ) CP = 2( S × S ) S × S CP ) == 2( S × S ) CP CP = ( S × S CP ) S × S CP ) CP == ( CP CP ) CP CP ) CP = 3 CP CP .An iteration of the usage of blow ups and blow downs (Theorems 9 and 10 ) allows usto conclude that the manifold 3 CP n CP admits a generalized complex structure. (cid:3) ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 13
For the remaining part of the section we proceed to consider more general pro-duction schemes of generalized complex structures on manifolds that are neithersymplectic nor complex, by taking a symplectic manifold as a starting point. Thefollowing constructions are motivated by [27].
Proposition 20.
Assume h ≥ . Let X be a simply connected symplectic4-manifold that contains a symplectic surface of genus two Σ ⊂ X that satisfies [Σ] = 0 , and π ( X − Σ) = 1 . The manifolds Z X,h := X h ( S × S ) and Z (cid:48) X,h := X h ( CP CP ) admit a generalized complex structure.Proof. We work out the case h = 1; the other cases follow verbatim from theargument. Consider the manifold Σ × T endowed with the product symplecticform. Let a , b , a , b be the generators of π (Σ ), and x, y the generators of π ( T ).It is easy to see that this manifold contains four pairs of homologically essentialLagrangian tori, and a symplectic surface of genus two. The tori are displayedbelow; the genus two surface is a parallel copy of the surface Σ × { pt } , and wewill continue to call it Σ during the proof. According to Baldridge and Kirk [7,Proposition 7], the fundamental group π (Σ × T − (Σ ∪ T ∪ · · · ∪ T ))is generated by the loops x, y, a , b , a , b . Moreover, with respect to certain pathsto the boundary of the tubular neighborhoods of the T i and Σ , the meridians andtwo Lagrangian push offs are given by • T : m = x, l = a , µ = [ b − , y − ], • T : m = y, l = b a b − , µ = [ x − , b ], • T : m = x, l = a , µ = [ b − , y − ], • T : m = y, l = b a b − , µ = [ x − , b ], • µ Σ = [ x, y ].The loops a , b , a , b lie on the genus 2 surface and form a standard set ofgenerators; [ a , b ][ a , b ] = 1 holds.Build the symplectic sum [14, Theorem 1.3] Z := X Σ=Σ Σ × T .Given that the loops a , b , a .b lie on Σ × { x } ⊂ Σ × T for x ∈ T , andthe meridian is given by µ Σ = [ x, y ], using van-Kampen’s Theorem we see thatour hypothesis π ( X − Σ) = 1 implies π ( Z ) = Z x ⊕ Z y . Perturb the symplecticform so that the tori { T , T , T , T } become symplectic [14, Lemma 1.6]. Performsimultaneously a (1 , , Z X, .By Theorem 8, the manifold Z X, admits a generalized complex structure, whichhas four type change loci. The surgeries set x = 1 = y . Thus, π ( Z X, ) = 1. Toconclude on the diffeomorphism type of Z X, , we observe the effect that the torussurgeries has on the embedded submanifolds. The tori y × b and y × a becomeembedded 2-spheres, each of which is a factor of an S × S summands. Thus, Z X, is diffeomorphic to X S × S ). In order to construct Z X,h for h ≥
2, one buildsthe symplectic sum Z := X Σ=Σ Σ × Σ h and submits it to surgical procedure similar to the one described above, making useof Proposition 15. The generalized complex manifold Z (cid:48) X,h is obtained by blowingup at a nondegenerate point on Z X,h (Theorem 9), and then blowing it down(Theorem 10). (cid:3)
To finalize this section, we produce generalized complex structures on more gen-eral connected sums.
Proposition 21.
Suppose h ≥ . Let X and Y be symplectic 4-manifolds thatcontain symplectic tori T X ⊂ X, T Y ⊂ Y such that [ T X ] = 0 = [ T Y ] , and π ( X ) = π ( X − T X ) = π ( Y ) = π ( Y − T Y ) = 1 . The manifolds Z X,Y, h − := X h − S × S ) Y and Z (cid:48) X,Y, h − := X h − CP CP ) Y admit a generalized complex structure. The following proof was inspired by an argument due to Z. Szab´o given in [27].
Proof.
Following Proposition 12, perturb the symplectic sum [14, Lemma 1.6] sothat the tori { T , T , T , T , T , T } become symplectic. Build the symplectic sum[14, Theorem 1.3] Z := X T X = T Σ × Σ T = T Y Y .Notice that in π ( Z ), we have a = c = a = c = 1. Simultaneously per-form (0 , , T , T , T , and T . Denote the resulting general-ized complex 4-manifold by Z X,Y, (Theorem 8). This manifold contains four typechange loci.Let us take a look at the surviving submanifolds of Z X,Y, . The torus surgeriesperformed along b and b turn the tori b × d , b × d , and the genus 2-surface { x } × Σ into imbedded 2-spheres of self-intersection zero. Each of these spheres isa factor of an S × S -summand. Therefore, Z X,Y, = X S × S ) Y .The generalized complex manifold constructed is neither symplectic [28] nor com-plex [22, 8]. The generalized complex manifold Z (cid:48) X,Y, = X CP CP ) Y isobtained by blowing up a nondegenerate point on Z X,h (Theorem 9), and thenblowing it down (Theorem 10). (cid:3)
Remark 22.
Submanifolds of higher genus.
The reader will notice that the sym-plectic sums involved in the previous arguments can be modified in order to yieldsimilar results in the case the starting manifold X contains symplectic submanifoldsof higher genus . Non-simply connected twisted generalized complex 4-manifolds
Given that our principal mechanism of construction is to apply torus surgeries tonon-simply connected building blocks, an organic next step is the study of existenceof twisted generalized complex structures in the non-simply connected realm.
ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 15
Finitely presented fundamental groups.
In his lovely paper [14], Gompfproved that every finitely presented group is the fundamental group of a symplectic4-manifold. Given that the existence of a symplectic structure naturally inducesthe existence of a generalized complex structure (Example 4), a posteriori the fun-damental group imposes no restriction on the existence of such a structure.Moreover, Kotschick [23] proved that any finitely presented group occurs as thefundamental group of an almost-complex 4-manifold. It is an interesting question,how much the size of a generalized complex 4-manifold depends on its fundamentalgroup. In this direction we obtain the following result.
Theorem 23.
Let G be a finitely presented group with a presentation consisting of g generators x , . . . , x g , and r relations w , . . . , w r , and let k be a nonnegative integer.There exists a non-symplectic twisted generalized complex 4-manifold X ( G, k ) with π ( X ( G, k )) ∼ = G , e ( X ( G, k )) = 4( g + r + 2) + k , and σ ( X ( G, k )) = − k . We point out that the manifolds of Theorem 23 do not share the homotopy typeof a complex surface [8]. The remaining part of the section is technical; see alsoRemark 24.In order to prove the theorem, we follow an argument due to Baldridge and Kirk.In [5], these authors build a symplectic manifold N , which serves as a fundamentalbuilding block for these constructions since it allows the manipulation on the num-ber of generators and relations on fundamental groups. We proceed to describe it.Begin with a 3-manifold Y that fibers over the circle, and build the 4-manifold N := Y × S .Its Euler characteristic and its signature are both zero, and it admits a symplecticstructure [5, p. 856].The fundamental group of N has the following presentation. π ( N ) = (cid:10) H, t (cid:12)(cid:12) R g ∗ ( x ) = txt − , x ∈ H (cid:11) × (cid:10) s (cid:11) ,where the group H has a presentation given by H = (cid:10) x , , y , , . . . , x g, , y g, , x , ,y , , . . . , x g, , y g, , . . . , x ,n , y ,n , . . . , x g,n , y g,n (cid:12)(cid:12) Π ni =1 Π gj =1 [ x j,i , y j,i ] (cid:11) .Let G be a finitely presented group with g generators and r relations. Thefundamental group π ( Y × S ) has classes s, t, γ , . . . , γ g + r so that G ∼ = π ( Y × S ) /N ( s, t, γ , . . . , γ g + r ),where N ( s, t, γ , . . . , γ g + r ) is the normal subgroup generated by the aforemen-tioned classes.The symplectic manifold N contains g + r + 1 symplectic tori T Y , T Y , . . . , T Yg + r ⊂ N of self-intersection zero that have two practical features regarding our fundamentalgroup computations. • The generators of π ( T Y ) represent s and t , and • the generators of π ( T Yi ) represent s and γ i . The curve s has the form { y } × S ⊂ Y × S , with y ∈ Y , and γ i has the form γ i × { x } ⊂ Y × { x } , x ∈ S . The role of the curves γ i is to provide the r relationsin a presentation of G [5, Section 4]; we point out that the number of curves γ i isgreater than r .Serious technical issues arise during cut-and-paste constructions in the computa-tion of fundamental groups, the choice of basepoint being the culprit. The argumentused here to prove Theorem 23 builds upon the careful analysis done by J. Yazinskiin [30, Section 4].Let R be the symplectic manifold obtained from Σ × Σ by applying the fol-lowing five Luttinger surgeries. Perform two (1 , , − T , T ,and three (0 , , −
1) -surgeries on the tori T , T , T . The manifold R contains thehomologically essential Lagrangian tori T , T , T ⊂ R . Perturb its symplectic formso that all these tori become symplectic [14, Lemma 1.6].Let us pin down a basepoint for the fundamental group computations that areto come. Set the basepoint to be v ∈ ∂ ( ν ( T )), where ν ( T ) is a tubular neigh-borhood of T . The fundamental group π ( R − T ) is generated by the elements { a , b , . . . , c , d , (cid:15) , (cid:15) , . . . , (cid:15) m } , where the elements { (cid:15) j } are conjugates to thebased meridian µ = [ b − , d − ]. Let β be a path inside R − T from the base-point ( x, y ) that was chosen to calculate π of the complement of the tori insideΣ × Σ (cf. [12] and Proposition 12) to v , which is contained in the torus b × d and such that it intersects T transversally at one point. The paths representinggenerators of π (Σ × Σ , ( x, y )) can be conjugated by β in order to obtain pathsthat are based at v . The relations stated in Proposition 12 continue to hold in π ( R − T , v ) under the new choice of basepoint. Proof.
The task at hand is to construct a twisted generalized complex manifold X ( G ) that is not symplectic, such that π ( X ( G )) = G, e ( X ( G )) = 4( g + r + 2), and σ ( X ( G )) = 0. This is done by applying torus surgeries to a symplectic sum alongtori composed of Y , a symplectic manifold built from Σ × Σ , and g + r + 1 copiesof R , which we will denote by R i , i = 1 , . . . , g + r . The symplectic tori inside eachcopy are denoted by T ,i , T ,i , T ,i ⊂ R i . Once X ( G ) is obtained, applying blowups to it results in the manifold X ( G, k ). The details are as follows.Consider the manifold Σ × Σ equipped with the product symplectic form. Asmall modification on the fundamental group computations in [21] and Proposition15 yields that the Lagrangian tori inside Σ × Σ , the meridians and the Lagrangianpushoffs of twelve tori are given by T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := a × c , m = a , l = c , µ = [ b − , d − ], T := b × c , m = b , l = d c d − , µ = [ a − , d ], T := b × c , m = b , l = d c d − , µ = [ a − , d ], ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 17 T := a × d , m = b a b − , l = d , µ = [ b , c − ], T := a × d , m = b a b − , l = d , µ = [ b , c − ], T := a × d , m = b a b − , l = d , µ = [ b , c − ], and T := a × d , m = b a b − , l = d , µ = [ b , c − ].Construct a symplectic manifold S by applying Luttinger surgeries and per-turbing the symplectic form as follows. Perform (1 , , − T , T , T , T , T , T , and on the tori T , T , T perform (0 , , − T , T ⊂ S become symplectic [14, Lemma 1.6]. Notice that H ( S ; Z ) ∼ = Z is generated by c .Regarding π ( S − T ), set the basepoint to be v ∈ ∂ ( ν ( T )). Let β be apath in S − ν ( T ) from the chosen basepoint ( x , y ) of π (Σ × Σ , ( x , y )) usedto compute the fundamental group of the complement of the tori to v , whichis contained in the torus b × c and intersects T transversally in one point.Conjugating the paths that represent generators of π (Σ × Σ , ( x , y )) by β yields paths based at v . The fundamental group π ( S − T ) is generated by theelements { a , b , . . . , c , d , (cid:15) S , . . . , (cid:15) Sm } , where the elements { (cid:15) Sj } are conjugates tothe based meridian µ .Following [14] and [5], we start by building a symplectic manifold that providesus with the generators of G (plus some generators that we will be getting ridof at a later stage with the usage of torus surgeries), and that contains enoughsubmanifolds to introduce the relations of G in a later step. For these purposes,build the symplectic sum [14, Theorem 1.3] Z := S T = T Y Y using the diffeomorphism φ : T → T Y that induces the identifications a (cid:55)→ s , d (cid:55)→ t on fundamental groups. The meridian µ Y is identified with the basedmeridian µ = [ b , c − ]. Let z be the basepoint for the block Z , and let η be apath that takes z to φ ( v ) with v ∈ ∂ ( ν ( T )). Conjugation by η provides uswith paths that represent elements in π . We abuse notation, and we keep callingthe conjugated elements with the same symbols.Using Seifert-van Kampen’s Theorem, the fundamental group is given by π ( Z ) = π ( Y − ν ( T Y )) ∗ π ( S − ν ( T )) .Here, ν ( T Y ) , ν ( T ) denote tubular neighborhoods of the corresponding tori; inparticular, these manifolds intersect in an open neighborhood of T .Moreover, the tori T Y , . . . , T Yr + g are contained in Z . The next step is to usethese tori to perform r + g symplectic sums with the purpose of introducing the r relations in the presentation of G . In the process, more relations are introducedinto the group presentation. We will get rid of any extra relation by using torussurgeries at a latter step. Take i = 1 , · · · , r + g copies of R , and call each copy R i . We start by choosing based curves in R i that are homotopic to the generators a ,i , and c ,i for all i . By Section 4.1, the based meridian µ ,i is taken to bethe commutator [ b − ,i , d − ,i ]. These push offs are contained in the boundary of thetubular neighborhood of the respective torus inside R i . Minding the choice of base point for the application of Seifert-van Kampen’sTheorem, we will follow the discussion that precedes the proof for each copy of R .So, we have basepoints v i ∈ ∂ ( ν ( T ,i )) and conjugating paths β i .Let Z (cid:48) := Z − ( ν ( T Y ) ∪ · · · ∪ ν ( T Yr + g )), where ν ( T Yi ) denotes a tubular neigh-borhood of the symplectic torus T Yi ⊂ Z . Build the symplectic sums of Z and R i along T ,i using the diffeomorphism φ i : T ,i → T Yi that identify the generators a ,i (cid:55)→ s i c ,i (cid:55)→ γ i to build a symplectic manifold ˆ X . The tori T ,i , T ,i , T (cid:48) are contained in ˆ X . Si-multaneously perform (0 , , , , , , c ,i = a ,i = c = 1. Denote the resulting twisted generalized complexmanifold by X ( G ). We claim that the fundamental group of X ( G ) is given by thefinitely presented group G of our hypothesis.We need to make sure that the usage of Seifert-van Kampen’s Theorem is donecarefully. There must be a common basepoint that is being shared by the openneighborhoods of the pieces being glued together. Set z to be the basepoint in Z (cid:48) . Regarding the basepoints v i ∈ ∂ ( ν ( T ,i )) for the pieces R i − T ,i , fix paths η i that take z to φ ( v ). Take R i − ν ( T ,i ) to be the union of R i − ν ( T ,i ) and aneighborhood of the path η i . We continue to abuse notation, and denote an elementof π ( R i − ν ( T ,i ) with the same symbol for the related element in π ( R i − ν ( T ,i )),since conjugation with the path η i correlates the elements and the relations continueto hold.By Seifert-van Kampen’s Theorem we have π ( X ( G )) ∼ = π ( Z (cid:48) ) ∗ π ( R − ν ( T , )) ∗···∗ π ( R r + g − ν ( T ,r + g )) N ,where N is the normal subgroup of π ( Y ) ∗ π ( S ) ∗ π ( R − T , ) ∗ · · · ∗ π ( R r + g − T ,r + g ) generated by a ,i s − i , c ,i γ − i ,the meridians µ (cid:48) µ Y − , µ Yi µ ,i − and theirconjugates, and c ,i = a ,i = c = 1. The element ¯ γ i is a based loop in Z (cid:48) thatis freely homotopic to a push off of the loop γ i × { x } ⊂ γ i × S to the boundaryof a tubular neighborhood of the torus T i = γ i × S . The element µ Yi is a basedmeridian corresponding o the torus T i ⊂ Z (cid:48) . Given that the curves ¯ s i and s areof the form { x } × S , the elements ¯ s i are based curves in Z (cid:48) that are homotopicto s on the boundary of a tubular neighborhood of T i = γ i × S (cf. [30, Proof ofLemma 4.3]).We need to see that the elements a , b , a , b , c , d , c , d , d − t, b ,i , b ,i , d ,i , d ,i (and the conjugates of the meridians (cid:15) ,i , . . . , (cid:15) n,i , (cid:15) S , . . . , (cid:15) Sm ) for i = 1 , . . . , g + r aretrivial in π ( X ( G )). The (1 , , , , b ,i , b ,i , d ,i , d ,i . This can be seen by plugging in any of the identities c ,i = 1 = a ,i in the presentation of π ( R i ) (see Proposition 12). Moreover, the triviality of thethree generators c ,i = a ,i = c = 1 implies that the meridians µ , µ Y , µ Yi , µ ,i andtheir conjugates are trivial in π ( X ( G )). Using the identity c = 1 in the relationsin π ( S ), we see that a = b = a = b = c = d = c = d = d = 1, and t = 1due to the identifications of the generators of the tori during the construction ofthe symplectic sum Z . Moreover, since µ Yi = 1 for i = 1 , . . . , r + g , their conjugatesare trivial. This implies that ¯ γ i is conjugate to γ i in π ( X ( G )). By the choice of ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 19 γ i [5], we have π ( X ( G )) ∼ = G .To argue that the manifold X ( G ) does not admit a symplectic structure, we pro-ceed as follows. The torus T (cid:48) is contained in X ( G ), since it was disjoint from all thetori involved in the surgeries. The (0 , , T (cid:48) turns T (cid:48) into an embedded 2-sphere of self-intersection zero. By the Adjunction inequality(Theorem 11), the manifold X ( G ) has trivial Seiberg-Witten invariants. Taubes’work [28] implies that X ( G ) does not admit a symplectic structure, as claimed.The computations of the characteristic numbers of X ( G ) are straight-forward. (cid:3) Remark 24.
Another construction of a manifold X ( G ) . A twisted generalizedcomplex 4-manifold that is neither symplectic, nor complex with arbitrary finitelypresented fundamental group, Euler characteristic 10 + 4( g + r + 1) and signature − X in [30, Theorem1.1] with Σ × Σ , and then apply accordingly six surgeries of Theorem 8 cf. Section4.1. Such a twisted generalized complex structure contains six type change locuscf. Remark 17. . Abelian groups.
It is of our interest to provide explicit constructions, whichwe hope will be useful in improving our understanding of twisted generalized com-plex structures. We begin this section with two examples that expose differenttwisted generalized complex structures on the same 4-manifold.
Example 25.
Twisted generalized complex structure on S × S with a single typechange locus (cf. [17, Example 6.38], [10, Example 4.1]). Consider T × S en-dowed with the product symplectic form. A (1 , , T × { s } for s ∈ S along one of the loops carrying a generator of π ( T × S ) results in S × S . The diffeomorphism is established as follows. Thehandlebody of T × S is given at the top of Figure 1. We are using the 1-handlenotation introduced by Akbulut in [1]. The effect that a (1 , , T × D is to interchange a 1-handle (dot-ted circle) and a 0-framed 2-handle [15, Figure 8.25, p.316], [2, Figure 17]. Thus,performing a (0 , , T × { pt } results in the second diagram ofFigure 1. The second and third rows show the isotopies and handle cancellationsthat establish a diffeomorphism with S × S . By Theorem 8, this primary Hopfsurface has a twisted generalized complex structure with one type change locus. Example 26.
Twisted generalized complex structure on blow-ups of L ( p, × S with one, two or three type change loci . To build a twisted generalized complexstructure on this secondary Hopf surface, start with the 4-torus T = T × T endowed with the product symplectic form. Denote by x, y, a, b both the generatorsof π ( T × T ) = Z x ⊕ Z y ⊕ Z a ⊕ Z b and the corresponding loops as well. Underthis notation, the tori T := x × a, T := y × a and their respective dual tori y × b, x × b are Lagrangian, and the torus T := a × b and its dual x × y are symplectic. Figure 1.
The result of performing a (1 , , T × S on the 2-torus T × { pt } is S × S According to Baldridge and Kirk [6, Section 2] and [7, Theorem 1], the funda-mental group of T − ( T ∪ T ) is generated by the loops x, y, a, b and the relations[ x, a ] = [ y, a ] = 1 hold. The meridians of the tori and the two Lagrangian push offsof their generators are given by the following formulae: m = x, l = a, µ = [ b − , y − ] and m = y, l = bab − , µ = [ x − , b ]. ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 21
Apply a ( p, , − T . This Luttinger surgery introduces therelation x p = [ b − , y − ]. Now, do a (1 , , T to introduce the relation y = [ x − , b ]. Last, perform a (0 , , T along b , and call the re-sulting manifold X p (1). The last surgery sets b = 1, which implies y = [ x − ,
1] = 1,and x p = 1. Since the elements x and a commute, we have π ( X p (1)) = < x, a : x p = 1 , [ x, a ] = 1 > ∼ = Z /p Z ⊕ Z .Theorem 8 implies that X p (1) admits a twisted generalized complex structurewith one single type change locus. If we simultaneously perform (1 , , T , and (0 , , T , a twisted generalized complexstructure ( X p (2) , H, J ) with two type change loci. Performing simultaneously thesurgery of Theorem 8 on three symplectic T , T , T yields a third twisted general-ized complex structure ( X p (3) , H, J ) with three type change loci.Set X p := X p ( i ) for i ∈ { , , } . Let us check that there exists a diffeomorphism X p → L ( p, × S , by using the fact that a diffeomorphism between two 3-manifoldsextends to a diffeomorphism between the 4-manifolds obtained by taking the prod-uct of S with the diffeomorphic 3-manifolds. We are grateful to Ronald J. Stern,having learned the following argument from him.Think of the 4-torus as T = T × S = ( x × y × b ) × a . The torus surgeriesperformed to obtained X p can be seen as Dehn surgeries surgeries on D × T =( D × S ) × S along the curves x, y, b in T (see Section 3.1). In particular, wehave that the twisted generalized complex manifold X p = (cid:102) T × S , where (cid:102) T is the3-manifold obtained from applying the three Dehn surgeries on the 3-torus.The diffeomorphism between (cid:102) T and L ( p,
1) can be seen through the analysis ofthe effect that the Dehn surgeries have on the handlebody of T . The 3-torus isobtained by 0-surgery on the Borromean rings [15, Figure 5.25, p. 159], and theLens spaces are obtained by performing a − p -surgery on the unknot [15, Figure5.24, p.158]. Thus, X p is diffeomorphic to L ( p, × S .By blowing up k non-degenerate points and iterating the usage of Theorem 9,one concludes that L ( p, × S k CP admits three twisted generalized complex structures with one, two, and three typechange loci. Remark 27.
New twisted generalized complex structures on blow-ups of S × S and T × S . Since there are diffeomorphisms L (1 ,
1) = S and L (0 ,
1) = S × S ,Example 26 covers the existence of twisted generalized complex structures on blow-ups of S × S and on blow ups of T × S as well. In particular, the twistedgeneralized complex structure on S × S has two type change loci. The twistedgeneralized complex structure on the Hopf surface displayed in Example 25 has asingle type change locus. Furthermore, it was proven above that there are twistedgeneralized complex structures on T × S with one or two type change loci . The main result of this section is the following theorem.
Theorem 28.
The manifolds • ˆ m ( S × S ) S × S , and • ˆ m CP n CP S × S admit a twisted generalized complex structure if and only if they admit an almostcomplex structure. The production of these manifolds involve torus surgeries on the product Σ × Σ g of a genus two surface and a genus g surface, as it was done for Theorem 16 inSection 4.2. In particular ˆ m = m + 1 = 2 g −
2. For the number of type change locisee Remark 17 and Section 4.1.
Proof.
The necessity of the existence of an almost-complex structure is clear [16].The construction of the manifolds claimed by the first item in the statement ofthe theorem was carried out in Proposition 18. To see that these manifolds are re-ducible, one proceeds as in the proof of Theorem 16. The existence of the manifoldsof the second claim in the statement follows by applying blow ups and blow downs(Theorem 9 and Theorem 10) to the manifolds of the first item (compare with theproof of Proposition 19). (cid:3)
The symmetric product of a genus g surface Sym (Σ g ) is a surface of general typeobtained by taking the quotient of the product of two surfaces of genus g Σ g × Σ g by the action of the involution Σ g × Σ g → Σ g × Σ g defined by ( x, y ) (cid:55)→ ( y, x ). Byapplying the procedure described in the previous sections to this K¨ahler manifoldone obtains the following proposition. Proposition 29.
Let k , k , k , k be a nonnegative integers. There are twistedgeneralized complex non-symplectic, non-complex 4-manifolds with for the followinghomeomorphism classes • π = 1 : ( g − g + 1) CP k CP , • π = Z /p Z : ( g − g + 1) CP k CP (cid:94) L ( p, × S • π = Z /p Z ⊕ Z /q Z : ( g − g + 1) CP k CP (cid:92) L ( p, × S • π = Z : ( g − g + 2) CP k CP S × S We finish the section by mentioning that it is straight-forward to obtain similarexistence results of twisted generalized complex manifolds that do not admit asymplectic nor a complex structure, and whose fundamental group is among thefollowing choices π = Z /p Z ⊕ Z /p Z ⊕ · · · ⊕ Z /p g − Z .6.3. Free and surface groups.
For what follows we start with the symplecticmanifold T × Σ g endowed with the product form. Theorem 30.
Let g ≥ , and assume k to be a nonnegative integer. There ex-ist twisted generalized complex non-symplectic 4-manifolds X F,g,k ( i ) and X S,g,k ( j ) ,which have e ( X F,g,k ) = e ( X S,g,k ) = k and σ ( X F,g,k ) = σ ( X S,g,k ) = − k such that • π ( X F,g,k ( i )) = g (cid:122) (cid:125)(cid:124) (cid:123) Z ∗ · · · ∗ Z • π ( X S,g,k ( j )) = π (Σ g ) .The twisted generalized complex structures on these manifolds can be chosen to havea number of i ∈ { , . . . , g } and j ∈ { , } type change loci respectively. ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 23
The starting point of the construction is the manifold T × Σ g , the product of a2-torus and a surface of genus g, equipped with the product symplectic form. Wegather the topological properties of this manifold that will be used in the followingresult. Lemma 31.
The first homology group H ( T × Σ g ; Z ) ∼ = Z g +2 is generated by x, y, a , b , · · · , a g , b g . The second homology H ( T × Σ g ; Z ) ∼ = Z g +2 is generatedby T × { s g } ( { s g } ∈ Σ g ) , { t } × Σ g ( { t } ∈ T ) , and the tori x × a i , y × a i , x × b i , y × b i where i = 1 , , · · · , g . The fundamental group π ( T × Σ g − (cid:83) gi =1 T i ) is generatedby the elements x, y, a , b , · · · , a g , b g , and the relations [ x, a i ] = [ y, a i ] = [ y, b i a i b − i ] = 1 for i = 1 , · · · , g , and [ x, y ] = 1 = [ a , b [ a , b ] · · · [ a g , b g ] hold in this group. The tori with the corresponding meridian, and Lagrangian pushoffs are given by • T : m = x, l = a , µ = [ b − , y − ] , • T : m = y, l = b a b − , µ = [ x − , b ] , • T : m = x, l = a , µ = [ b − , y − ] , • T : m = y, l = b a b − , µ = [ x − , b ] , • T : m = x, l = a , µ = [ b − , y − ] ,... • T g − : m g − = x, l g − = a g − , µ g − = [ b − g , y − ] , and • T g : m g = y, l g = b g a g b − g , µ g = [ x − , b g ] . The proof of this lemma is omitted. It is an straight-forward generalization of [7,Proposition 7]; compare with Proposition 15. The proof for both instances of thetheorem consists of building a generalized complex 4-manifold with trivial Eulercharacteristic, trivial signature, and with the desired fundamental group. Then oneconcludes by blowing up k points on such a manifold (Theorem 9). Let us startwith the case of surface groups. Proof. (Theorem 22). Perturb the product symplectic form of T × Σ g [14, Lemma1.6] so that T and T become symplectic. Denote by X S,g (2) the twisted generalizedcomplex 4-manifold obtained from T × Σ g by applying (1 , , T and on T . Theorem 8 implies that X S,g (2) admits a twisted generalized complexstructure with two type change loci. We have e ( X S,g (2)) = 0 = σ ( X S,g (2)). Indeed,the Euler characteristic and the signature remain invariant under torus surgeries,so e ( X S,g (2)) = e ( T × Σ g ) = 0 and σ ( X S,g (2)) = e ( T × Σ g ) = 0. Moreover, thesurgeries set x = 1 = y . Thus, π ( X S,g (2)) = π (Σ g ).The manifold X S,g,k (2) is a twisted generalized complex 4-manifold obtained byblowing up X S,g (2) at k non-degenerate points (Theorems 8 and 9). A straight-forward computation yields e ( X S,g,k (2)) = k, σ ( X S,g,k (2)) = − k . Since blow upsdo not alter the fundamental group, we have π ( X S,g,k (2)) = π (Σ g ). The twistedgeneralized complex structure with one type change locus is built by doing (1 , , T , and a (1 , , T .Let us consider the case of free fundamental groups of rank g . We claim thatthere exists a twisted generalized complex 4-manifold X F,g ( i ) with i type change loci, with trivial Euler characteristic, trivial signature, and whose fundamentalgroup is generated by b , b , . . . , b g so that π ( X F,g ( i )) = Z b ∗ · · · ∗ Z b g .The manifold X F,g,k ( i ) is obtained by blowing up X F,g ( i ) at k non-degeneratepoints (Theorem 9). As in the previous case, by a straight-forward computation,one checks e ( X F,g,k ( i )) = k, σ ( X F,g,k ( i )) = − k , and π ( X F,g,k ) = g (cid:122) (cid:125)(cid:124) (cid:123) Z ∗ · · · ∗ Z .The twisted generalized complex manifold X F,g ( i ) is constructed as follows. Wework out the case of one single type change locus (i = 1); the other cases followfrom a small variation of the argument as in the surface groups case that wasdiscussed before. Perturb the symplectic form on T × Σ g so that the torus T becomes symplectic. Perform (0 , , − T , (1 , , − T ,and (0 , , − T j for 4 ≤ j ≤ g + 2. Peform (1 , , T .The relation induced by the last surgery is y = 1; by using this on the relationsintroduced by the Luttinger surgeries, one sees x = a = a = · · · = a g = 1. Thisimplies π ( X F,g (1)) = Z b ∗ · · · ∗ Z b g .By Theorem 8, X F,g (1) is a twisted generalized complex manifold, and it has trivialEuler characteristic and signature zero.The triviality of the Seiberg-Witten invariants of any manifold constructed abovefollows from Theorem 11, by keeping track of the submanifolds on Lemma 31 andobserving the existence of an imbedded 2-sphere of self-intersection zero. (cid:3) On further research
We finish the paper with two interesting questions.
Question 32.
What are the sufficient conditions for a manifold to admit a gener-alized complex structure in dimension four?
Question 33.
What is the relation between an almost-complex structure and ageneralized complex structure on a given smooth 4-manifold? acknowledgments The author is indebted to Gil Cavalcanti, to Marco Gualtieri and to Nigel Hitchinfor fruitful conversations that led to the writing of the present paper. Special thanksare due to Jonathan Yazinski for useful comments on an earlier draft. We thankRonald Fintushel, Robert E. Gompf, Paul Kirk, Jonathan Normand, and RonaldJ. Stern for helpful e-mail exchanges, useful discussions, and/or for their patienceanswering questions. Their expertise was of significant value in the production ofthe manuscript. We thank Utrecht University and University of Toronto for theirhospitality during the production of the paper.The Simons Foundation is gratefully acknowledged for the support under whichthis work was carried out.
ENERALIZED COMPLEX STRUCTURES IN DIMENSION 4 25
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