A new symplectic surgery: the 3-fold sum

Symington, Margaret

doi:10.1016/s0166-8641(97)00197-1

Cited by 12 publications

(37 citation statements)

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“…In [14], Symington shows that if all manifolds are symplectic and the surfaces are symplectically embedded then one can arrange for the 3-fold sum to be symplectic, and similarly in the case of the next proposition.…”

Section: Pinwheelsmentioning

confidence: 94%

“…In her thesis (cf [14]), Symington discussed the operation of symplectic summing 4-manifolds along surfaces embedded with normal crossings -the k -fold sum. We study this operation now from a topological point of view.…”

Section: Pinwheelsmentioning

confidence: 99%

“…The next example exhibits CP 2 as a 3-component pinwheel made up of three 4-balls (the standard coordinate neighborhoods in CP 2 ). Our description is essentially that of Symington [14], using the language of Orlik and Raymond [13] rather than that of toric varieties. Example This example is described by Figure 9.…”

Section: Consider the Projectionmentioning

confidence: 99%

“…It now follows from [14] that Q 3 is a symplectic manifold with a symplectic structure extending those of the pinwheel components.…”

Section: Propositionmentioning

confidence: 99%

“…More specifically, a pinwheel structure is a generalization of the idea of a k -fold symplectic sum which was introduced by Symington [14]. The basic idea is this: One has a sequence f.X i I S i ; T i /g of 4-manifolds X i with embedded surfaces S i , T i , which intersect transversely at a single point and with the genus g.T i / D g.S iC1 /.…”

Section: Introductionmentioning

confidence: 99%

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Pinwheels and nullhomologous surgery on 4–manifolds withb⁺= 1

Fintushel

Stern

2011

Algebr. Geom. Topol.

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1649Pinwheels and nullhomologous surgery on 4-manifolds with b C D 1 RONALD FINTUSHEL RONALD J STERNWe present a method for finding embedded nullhomologous tori in standard 4-manifolds which can be utilized to change their smooth structure. As an application, we show how to obtain infinite families of simply connected smooth 4-manifolds with b C D 1 and b D 2; : : : ; 7, via surgery on nullhomologous tori embedded in the standard manifolds CP 2 #k CP

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Section: Pinwheelsmentioning

confidence: 94%

Section: Pinwheelsmentioning

confidence: 99%

Section: Consider the Projectionmentioning

confidence: 99%

“…It now follows from [14] that Q 3 is a symplectic manifold with a symplectic structure extending those of the pinwheel components.…”

Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pinwheels and nullhomologous surgery on 4–manifolds withb⁺= 1

Fintushel

Stern

2011

Algebr. Geom. Topol.

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GW invariants relative to normal crossing divisors

Ionel

2015

Advances in Mathematics

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In this paper we introduce a notion of symplectic normal crossing divisor V and define the GW invariant of a symplectic manifold X relative to such a divisor. Our definition includes normal crossing divisors from algebraic geometry. The invariants we define in this paper are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [16], which covered the case V was smooth. The main step is the construction of a compact moduli space of relatively stable maps into the pair (X, V ) in the case V is a symplectic normal crossing divisor in X.In previous work with Thomas H. Parker [16] we constructed the relative GromovWitten invariant GW (X, V ) of a closed symplectic manifold X relative to a smooth "divisor" V , that is a (real) codimension 2 symplectic submanifold. These relative invariants are defined by choosing an almost complex structure J on X that is compatible with both V and the symplectic form, and counting J-holomorphic maps that intersect V with specified multiplicities. An important application is the symplectic sum formula ✩ maps, or those that are (C * ) m -equivariant, and all come with corresponding Gromov-type extensions J V. The restriction to level zero induces a fibrationand the rescaling process defines a family of sections (ω ε , J) in it. So J V(X m , V m ) can also be thought as an extension of the parameter space J (X, V ) indexing deformations of Eq. (2.5) for a map f : C → X m , giving rise to a universal moduli space M(X m , V m ) → J V(X m , V m ) consisting of maps into X m satisfying the refined matching conditions (7.6) along the singular locus. As mentioned in Remark 4.19, these parameter spaces come in various flavors, depending on how much of the structure of X m → X they are required to preserve. When restricted to the subspace of parameters which are invariant under the (C * ) m action on the positive levels, we also get a corresponding 'rubber' moduli space M τ (X m , D m )/T τ that appears in (7.9) and which describes the level m stratum of the relative moduli space M(X, V ).Remark A.7. When using Banach norms on the parameter spaces J V (rather than the Frechet topology C ∞ ), the process of rescaling loses m derivatives: e.g. the 'section' aboveHowever, for each s we have an a priori topological bound on the maximum number M of levels entering into the compactification M s (X, V ). So for the purpose of transversality we can start with Sobolev norms W l+M,p on the parameter space J (X, V ) and then take the limit as l → ∞ to get a smooth model for the resolution of each stratum of M s (X, V ).For a family X → B of deformations of a level m building, there is a space J (X → B) of parameters compatible with both the fibration and the total divisor. Since p is a singular fibration, this means that their restriction over each open stratum of B (over which p is a fibration) is compatible with p in the sense of (A.27). The compatibility with the total divisor means that they lift to a pair of parameters ...

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Symplectic surgeries along certain singularities and new Lefschetz fibrations

Akhmedov

Katzarkov

2020

Advances in Mathematics

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We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations (ta 0 ta 1 ta 2 · · · ta 2g+1 ) 2g+1 = t b 1 t g b 2 t b 3 , also known as the trident relations, in the mapping class group Γg,3 of an orientable surface of genus g ≥ 1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words (tc 1 tc 2 · · · tc 2g−1 tc 2g t 2 c 2g+1 tc 2g tc 2g−1 · · · tc 2 tc 1 ) 2n = 1, (tc 1 tc 2 · · · tc 2g tc 2g+1 ) (2g+2)n = 1 and (tc 1 tc 2 · · · tc 2g−1 tc 2g ) 2(2g+1)n = 1 in the mapping class group Γg of the closed orientable surface of genus g ≥ 1 and n ≥ 1. Furthemore, we show that the total spaces of some of these Lefschetz fibraions are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S 2 with non-separating vanishing cycles.

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A new symplectic surgery: the 3-fold sum

Cited by 12 publications

References 20 publications

Pinwheels and nullhomologous surgery on 4–manifolds withb⁺= 1

Pinwheels and nullhomologous surgery on 4–manifolds withb⁺= 1

GW invariants relative to normal crossing divisors

Symplectic surgeries along certain singularities and new Lefschetz fibrations

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A new symplectic surgery: the 3-fold sum

Cited by 12 publications

References 20 publications

Pinwheels and nullhomologous surgery on 4–manifolds withb+= 1

Pinwheels and nullhomologous surgery on 4–manifolds withb+= 1

GW invariants relative to normal crossing divisors

Symplectic surgeries along certain singularities and new Lefschetz fibrations

Contact Info

Product

Resources

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Pinwheels and nullhomologous surgery on 4–manifolds withb⁺= 1

Pinwheels and nullhomologous surgery on 4–manifolds withb⁺= 1