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Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
Abstract

Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
Symplectic genus, minimal genus and diffeomorphisms
"... 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 symp ..."
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Cited by 9 (4 self)
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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. Let M be a smooth, closed oriented 4−manifold. An orientationcompatible 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 CM is defined as the set of cohomology classes which are represented by orientationcompatible
Minimal genus problem: New approach Abstract
"... The minimal genus problem of connected sums of 4manifolds and the minimal slice genus of knots in CP 2 are treated. The approach used is twisting operations on knots in S 3. We give an upper bound of the smooth slice genus of lefthanded torus knots in CP 2 and we study the smooth slice genus of th ..."
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The minimal genus problem of connected sums of 4manifolds and the minimal slice genus of knots in CP 2 are treated. The approach used is twisting operations on knots in S 3. We give an upper bound of the smooth slice genus of lefthanded torus knots in CP 2 and we study the smooth slice genus of the family of (2, q)torus knots in CP 2 for any q ≥ 3. T. Lawson conjectured in [23] that the minimal genus of (m, n) ∈ H2(CP 2 #CP 2) is given by ( m−1 2) + ( n−1 2)this is the genus realized by the connected sum of algebraic curves in each factor. T. Lawson also conjectured in [23] that if X = X1#X2 is the connected sum of two symplectic 4manifolds with b + 2 ≥ 3, and if (a, b) ∈ H2(X) = H2(X1)⊕H2(X2) satisfies a.a ≥ 0 and b.b ≥ 0, then the minimal genus for this class is the sum of the minimal genus for the class a and the minimal genus for the class b. We answer these conjectures by the negative.