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252
Holomorphic triangles and invariants for smooth fourmanifolds
"... Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute gradi ..."
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Cited by 124 (24 self)
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Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute
On Park’s exotic smooth four manifolds
"... Abstract. In a recent paper, Park constructs certain exotic simplyconnected fourmanifolds with small Euler characteristics. Our aim here is to prove that the fourmanifolds in his constructions are minimal. 1. ..."
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Cited by 5 (1 self)
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Abstract. In a recent paper, Park constructs certain exotic simplyconnected fourmanifolds with small Euler characteristics. Our aim here is to prove that the fourmanifolds in his constructions are minimal. 1.
On irreducible fourmanifolds
"... For many years, four–manifold folklore suggested that all simply connected smooth four–manifolds should be connected sums of complex algebraic surfaces, with both their complex and non–complex orientations allowed 1. The first counterexamples were constructed in 1990 by ..."
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Cited by 6 (0 self)
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For many years, four–manifold folklore suggested that all simply connected smooth four–manifolds should be connected sums of complex algebraic surfaces, with both their complex and non–complex orientations allowed 1. The first counterexamples were constructed in 1990 by
Monopole classes and Perelman’s invariant of fourmanifolds
, 2006
"... We calculate Perelman’s invariant for compact complex surfaces and a few other smooth fourmanifolds. We also prove some results concerning the dependence of Perelman’s invariant on the smooth structure. ..."
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Cited by 6 (0 self)
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We calculate Perelman’s invariant for compact complex surfaces and a few other smooth fourmanifolds. We also prove some results concerning the dependence of Perelman’s invariant on the smooth structure.
Dissolving fourmanifolds and positive scalar curvature
 Math. Z
"... ABSTRACT. We prove that many simply connected symplectic fourmanifolds dissolve after connected sum with only one copy of S 2 × S 2. For any finite group G that acts freely on the threesphere we construct closed smooth fourmanifolds with fundamental group G which do not admit metrics of positive s ..."
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Cited by 7 (2 self)
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ABSTRACT. We prove that many simply connected symplectic fourmanifolds dissolve after connected sum with only one copy of S 2 × S 2. For any finite group G that acts freely on the threesphere we construct closed smooth fourmanifolds with fundamental group G which do not admit metrics of positive
Holomorphic triangle invariants and the topology of symplectic fourmanifolds
 Duke Math. J
"... This article analyzes the interplay between symplectic geometry in dimension 4 and the invariants for smooth fourmanifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic fourmanifolds, which leads to new ..."
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Cited by 46 (5 self)
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This article analyzes the interplay between symplectic geometry in dimension 4 and the invariants for smooth fourmanifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic fourmanifolds, which leads
Lagrangian matching invariants for fibred fourmanifolds: I
, 2008
"... In a pair of papers, we construct invariants for smooth fourmanifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the DonaldsonSmith invariant for Lefschetz fibrations. The ‘Lagrangian matching invariants ’ are designed to be ..."
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Cited by 39 (5 self)
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In a pair of papers, we construct invariants for smooth fourmanifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the DonaldsonSmith invariant for Lefschetz fibrations. The ‘Lagrangian matching invariants ’ are designed
FOURMANIFOLDS WITHOUT EINSTEIN METRICS
 MATHEMATICAL RESEARCH LETTERS 3, 133–147 (1996)
, 1996
"... It is shown that there are infinitely many compact simply connected smooth 4manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality 2χ>3τ. The examples in question arise as nonminimal complex algebraic surfaces of general type, and the meth ..."
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Cited by 77 (14 self)
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It is shown that there are infinitely many compact simply connected smooth 4manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality 2χ>3τ. The examples in question arise as nonminimal complex algebraic surfaces of general type
Absolutely graded Floer homologies and intersection forms for fourmanifolds with boundary
 Advances in Mathematics 173
, 2003
"... Abstract. In [22], we introduced absolute gradings on the threemanifold invariants developed in [21] and [20]. Coupled with the surgery long exact sequences, we obtain a number of three and fourdimensional applications of this absolute grading including strengthenings of the “complexity bounds ” ..."
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Cited by 183 (28 self)
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” derived in [20], restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF + for a variety of threemanifolds. Moreover, we show how the structure of HF + constrains the exoticness of definite intersection forms for smooth fourmanifolds which bound a given threemanifold
Singular connection and Riemann theta function
, 1997
"... We prove the ChernWeil formula for SU(n + 1)singular connections over the complement of an embedded oriented surface in smooth four manifolds. The expression of the representation of a number as a sum of nonvanishing squares is given in terms of the representations of a number as a sum of squares. ..."
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We prove the ChernWeil formula for SU(n + 1)singular connections over the complement of an embedded oriented surface in smooth four manifolds. The expression of the representation of a number as a sum of nonvanishing squares is given in terms of the representations of a number as a sum of squares
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