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72
INTEGRATION OVER THE uPLANE IN DONALDSON THEORY
, 1997
"... We analyze the uplane contribution to Donaldson invariants of a fourmanifold X. For b + 2(X)> 1, this contribution vanishes, but for b + 2 =1, the Donaldson invariants must be written as the sum of a uplane integral and an SW contribution. The uplane integrals are quite intricate, but can be anal ..."
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Cited by 52 (2 self)
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We analyze the uplane contribution to Donaldson invariants of a fourmanifold X. For b + 2(X)> 1, this contribution vanishes, but for b + 2 =1, the Donaldson invariants must be written as the sum of a uplane integral and an SW contribution. The uplane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the uplane integrals, the relation of Donaldson theory to N = 2 supersymmetric YangMills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to fourmanifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields.
Symplectic convexity in lowdimensional topology
 Proceedings of the Georgia Topology Conference
, 1998
"... Abstract. In this paper we will survey the the various forms of convexity in symplectic geometry, paying particular attention to applications of convexity inlow dimensional topology. ..."
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Cited by 29 (7 self)
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Abstract. In this paper we will survey the the various forms of convexity in symplectic geometry, paying particular attention to applications of convexity inlow dimensional topology.
Symplectic Fillings and Positive Scalar Curvature
 GEOM. AND TOPOLOGY
, 1998
"... Let X be a 4manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b^+_ 2 (X) > 0 or the bou ..."
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Cited by 26 (9 self)
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Let X be a 4manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b^+_ 2 (X) > 0 or the boundary of X is disconnected. As an application we show that the Poincaré homology 3sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3manifold which is not symplectically semifillable. Using work of Fr yshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3spheres having positive scalar curvature metrics.
Product formulas along T 3 for SeibergWitten invariants
 Math. Res. Letters
, 1997
"... Suppose that X is a smooth closed oriented 4manifold, and that X contains a smoothly embedded 2torus T 2 ↩ → X with trivial selfintersection number. Similarly to Dehnsurgery on knots in 3manifolds, a generalized logarithmic ..."
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Cited by 26 (1 self)
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Suppose that X is a smooth closed oriented 4manifold, and that X contains a smoothly embedded 2torus T 2 ↩ → X with trivial selfintersection number. Similarly to Dehnsurgery on knots in 3manifolds, a generalized logarithmic
On the moduli space of diffeomorphic algebraic surfaces
 Invent. Math
"... Abstract. In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented fourmanifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic descripti ..."
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Cited by 25 (2 self)
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Abstract. In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented fourmanifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic description of rational blow down, classical BrillNoether theory and deformation theory of normal flat abelian covers. One of the main problems concerning the differential topology of algebraic surfaces leaving unsolved by the “SeibergWitten revolution ” was to determine whether the differential type of a compact complex surface determines the deformation type. Two compact complex manifolds have the same deformation type if they are fibres of a proper smooth family over a connected
Minimal entropy and collapsing with curvature bounded from below
 Invent. Math
"... Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting ..."
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Cited by 25 (4 self)
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Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting an Fstructure is zero. We also show that if M admits an Fstructure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is nonnegative. We show that Fstructures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, CP 2, CP 2, S 2 × S 2 and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4, CP 2, S 2 × S 2, CP 2 #CP 2 or CP 2 #CP 2. Finally, suppose that M is a closed simply connected 5manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5, S 3 ×S 2, the nontrivial S 3bundle over S 2 or the Wumanifold SU(3)/SO(3). 1.
Higher Type Adjunction Inequalities In SeibergWitten Theory
, 1998
"... In this paper, we derive new adjunction inequalities for embedded surfaces with nonnegative selfintersection number in fourmanifolds. These formulas are proved by using relations between SeibergWitten invariants which are induced from embedded surfaces. To prove these relations, we develop the r ..."
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Cited by 23 (5 self)
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In this paper, we derive new adjunction inequalities for embedded surfaces with nonnegative selfintersection number in fourmanifolds. These formulas are proved by using relations between SeibergWitten invariants which are induced from embedded surfaces. To prove these relations, we develop the relevant parts of a Floer theory for fourmanifolds which bound circlebundles over Riemann surfaces. 1.
Seiberg–Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers
 Turkish J. of Math
, 1999
"... Let f: Σ → Σ be an orientation preserving diffeomorphism of a compact oriented Riemann surface. This paper relates the SeibergWitten invariants of the mapping torus Yf to the Lefschetz invariants of f. 1 ..."
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Cited by 20 (3 self)
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Let f: Σ → Σ be an orientation preserving diffeomorphism of a compact oriented Riemann surface. This paper relates the SeibergWitten invariants of the mapping torus Yf to the Lefschetz invariants of f. 1