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The genus of embedded surfaces in the projective plane
 Math. Research Letters
, 1994
"... Abstract. We show how the new invariants of 4manifolds resulting from the SeibergWitten monopole equation lead quickly to a proof of the ‘Thom conjecture’. 1. Statement of the result The genus of a smooth algebraic curve of degree d in CP 2 is given by the formula g =(d−1)(d − 2)/2. A conjecture s ..."
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Cited by 87 (1 self)
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Abstract. We show how the new invariants of 4manifolds resulting from the SeibergWitten monopole equation lead quickly to a proof of the ‘Thom conjecture’. 1. Statement of the result The genus of a smooth algebraic curve of degree d in CP 2 is given by the formula g =(d−1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4andford = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,8,10]. In this note we confirm the conjecture. Theorem 1. Let Σ be an oriented 2manifold smoothly embedded in CP 2 so as to represent the same homology class as an algebraic curve of degree d. Then the genus g of Σ satisfies g ≥ (d − 1)(d − 2)/2. Very recently, Seiberg and Witten [11,12,13] introduced new invariants of 4manifolds, closely related to Donaldson’s polynomial invariants [2], but in many respects much simpler to work with. The new techniques have led to more elementary proofs of many theorems in the area. Given the monopole equation and the vanishing theorem which holds when the scalar curvature is positive (something which was pointed out by Witten), the rest of the argument presented here is not hard to come by. A slightly different proof of the Theorem, based on the same techniques, has been found by Morgan, Szabo and Taubes. It is also possible to prove a version of Theorem 1 for other complex surfaces, without much additional work. This and various other applications will be treated in a later paper, with joint authors. 2. The monopole equation and the SeibergWitten invariants Let X be an oriented, closed Riemannian 4manifold. Let a spin c structure on X be given. We write c for the spin c structure and write W + = W + c and W − = W − c for the associated spin c bundles. Thus W + is a U(2) bundle and Clifford multiplication
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 ..."
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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
STABILISATION, BORDISM AND EMBEDDED SPHERES
, 2000
"... Abstract. It is one of the most interesting facts in 4–dimensional topology that even in simply–connected 4–manifolds, not every homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that many of the obstructions against constructing such a sph ..."
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Cited by 1 (1 self)
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Abstract. It is one of the most interesting facts in 4–dimensional topology that even in simply–connected 4–manifolds, not every homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that many of the obstructions against constructing such a sphere vanish if one modifies the ambient 4–manifold by adding copies of products of spheres, a process which is usually called stabilisation. In this paper, we extend this result to non–simply connected 4–manifolds and show how it is related to the Spin c –bordism groups of Eilenberg–McLane spaces.
STABILISATION, BORDISM AND EMBEDDED SPHERES IN 4–MANIFOLDS
, 2001
"... Abstract. It is one of the most important facts in 4–dimensional topology that there are 4–manifolds in which not every spherical homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions ..."
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Cited by 1 (0 self)
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Abstract. It is one of the most important facts in 4–dimensional topology that there are 4–manifolds in which not every spherical homology class of degree 2 can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4–manifold by adding products of 2–spheres, a process which is usually called stabilisation. In this paper, we extend this result to non–simply connected 4–manifolds and show how it is related to the Spin c – bordism groups of Eilenberg–McLane spaces.
Christian Bohr
, 2002
"... Abstract It is one of the most important facts in 4–dimensional topology that not every spherical homology class of a 4–manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a spher ..."
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Abstract It is one of the most important facts in 4–dimensional topology that not every spherical homology class of a 4–manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4–manifold by adding products of 2–spheres, a process which is usually called stabilisation. In this paper, we extend this result to non–simply connected 4–manifolds and show how it is related to the Spin c –bordism groups of Eilenberg–MacLane spaces. AMS Classification 57M99; 55N22
IMMERSIONS OF SURFACES IN ALMOST–COMPLEX 4–MANIFOLDS
, 2000
"... Abstract. In this note, we investigate the relation between double points and complex points of immersed surfaces in almost–complex 4–manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points ..."
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Abstract. In this note, we investigate the relation between double points and complex points of immersed surfaces in almost–complex 4–manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provide a generalization of a classical genus estimate due to V.A. Rokhlin to the case of immersed surfaces. 1.