Minimal genus embeddings in $S^2-$bundles over surfaces

Li, Bang He; Li, Tian Jun

doi:10.4310/mrl.1997.v4.n3.a7

Cited by 8 publications

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“…In light of the SW dimension calculation above and wall crossing formulas (12) and (13), Theorem 1.4 follows from the following more general result which involves the wall crossing value of the full SW. Theorem 3.3. Let X be a smooth, closed, connected, oriented 4-manifold with b + (X) = 1, and Σ ⊂ X a connected, smooth, embedded surface with Σ · Σ ≥ 0.…”

Section: Two Types Of Invariants For Type I Adjunction Classes We Nmentioning

confidence: 99%

“…and F is a generator for ImT . Formula (12) follows from [7,14], and the first equality in (13) follows from [23,Theorem 16] (see also [7] and [22]). The second equality in (13) follows from direct computation.…”

Section: Two Types Of Invariants For Type I Adjunction Classes We Nmentioning

confidence: 99%

“…Proof. The fact that D(X) = Aut(Γ) T for S 2 × S 2 , S 2 -bundles over T 2 is straightforward (see e.g [12] Remark 3). The equality for CP 2 ♯nCP 2 , n ≤ 9 was due to Wall [29] Theorem 2 and the following corollary, and for Enriques surface due to Lönne [18] Theorem 8 (extending Friedman-Morgan [2]).…”

Section: Applicationsmentioning

confidence: 99%

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Minimal genus for 4-manifolds withb⁺ = 1

Dai

Ho²,

2015

J Topology

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Section: Two Types Of Invariants For Type I Adjunction Classes We Nmentioning

confidence: 99%

Section: Two Types Of Invariants For Type I Adjunction Classes We Nmentioning

confidence: 99%

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Minimal genus for 4-manifolds withb⁺ = 1

Dai

Ho²,

2015

J Topology

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“…It therefore became an interesting question to find for a given homology class in a 4-manifold the minimal genus of an embedded closed connected oriented surface realizing that class. This question has been solved at least partly for rational and ruled surfaces and for 4-manifolds with a free circle action [4,5,14,15,16,17,18,19,25]. On symplectic 4-manifolds the question is related to the Thom conjecture [12,22,23].…”

Section: Introductionmentioning

confidence: 99%

The minimal genus problem for elliptic surfaces

Hamilton

2014

Isr. J. Math.

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“…Such manifolds include the Enriques surface, hyperelliptic surfaces, any torus bundle over torus which has b + = 1. In addition, from the results in [LiL4], [Li1] and [Kr2], manifolds with the property that two genera coincide for any class of positive square include minimal irrational ruled manifold, rational manifold with b − ≤ 9 and the product of a circle with a fibered 3−manifold Y with b 1 (Y ) = 1.…”

Section: Propertymentioning

confidence: 99%

Symplectic genus, minimal genus and diffeomorphisms

Li¹,

Li²

2002

Asian Journal of Mathematics

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Abstract. In this paper, the symplectic genus for any 2−dimensional class in a 4−manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least −1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least −1 in any 4−manifold admitting a symplectic structure. §1 Introduction Let M be a smooth, closed oriented 4−manifold. An orientation-compatible symplectic form on M is a closed two−form ω such that ω ∧ ω is nowhere vanishing and agrees with the orientation. For any oriented 4−manifold M , its symplectic cone C M is defined as the set of cohomology classes which are represented by orientationcompatible symplectic forms. For any class e ∈ H 2 (M ; Z), its minimal genus m(e) is the minimal genus of a smoothly embedded connected surface representing the Poincaré dual PD(e). The problem of determining the minimal genus has involved many of the important techniques in 4−manifold topology, and it bears its origin in the older problem of representing the Poincaré dual to a class by an embedded sphere (See the excellent survey papers [La1-2] and [Kr1] on these two problems).We are here interested in studying both these problems for 4−manifolds with non-empty symplectic cone. We will introduce the notion of symplectic genus η(e) for 4−manifolds with non-empty symplectic cone. Recall that any symplectic structure ω determines a homotopy class of compatible almost complex structures on the cotangent bundle, whose first Chern class is called the canonical class of ω. Roughly, the symplectic genus η(e) of a class e is given by the formula [e 2 +K · e]/2 +1, where K has largest pairing against e amongst canonical classes of symplectic structures for which the symplectic area of e is positive.η(e) has many nice properties, among which are (i) invariance under the action 1

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Minimal genus embeddings in $S^2-$bundles over surfaces

Abstract: Abstract. In this paper we completely solve the problem of minimal genus smooth embeddings for the S 2 −bundles over surfaces.

Cited by 8 publications

References 0 publications

Minimal genus for 4-manifolds withb⁺ = 1

Minimal genus for 4-manifolds withb⁺ = 1

The minimal genus problem for elliptic surfaces

Symplectic genus, minimal genus and diffeomorphisms

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Minimal genus embeddings in $S^2-$bundles over surfaces

Abstract: Abstract. In this paper we completely solve the problem of minimal genus smooth embeddings for the S 2 −bundles over surfaces.

Cited by 8 publications

References 0 publications

Minimal genus for 4-manifolds withb+ = 1

Minimal genus for 4-manifolds withb+ = 1

The minimal genus problem for elliptic surfaces

Symplectic genus, minimal genus and diffeomorphisms

Contact Info

Product

Resources

About

Minimal genus for 4-manifolds withb⁺ = 1

Minimal genus for 4-manifolds withb⁺ = 1