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252
Holomorphic triangles and invariants for smooth four-manifolds
"... Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in [8] and [12]. This four-dimensional theory also endows the corresponding three-dimensional 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 four-manifolds, built using the Floer homology theories defined in [8] and [12]. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute
On Park’s exotic smooth four manifolds
"... Abstract. In a recent paper, Park constructs certain exotic simply-connected 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 simply-connected fourmanifolds with small Euler characteristics. Our aim here is to prove that the fourmanifolds in his constructions are minimal. 1.
On irreducible four-manifolds
"... 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 four-manifolds
, 2006
"... We calculate Perelman’s invariant for compact complex surfaces and a few other smooth four-manifolds. 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 four-manifolds. We also prove some results concerning the dependence of Perelman’s invariant on the smooth structure.
Dissolving four-manifolds 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 three-sphere we construct closed smooth four-manifolds 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 three-sphere we construct closed smooth four-manifolds with fundamental group G which do not admit metrics of positive
Holomorphic triangle invariants and the topology of symplectic four-manifolds
- Duke Math. J
"... This article analyzes the interplay between symplectic geometry in dimension 4 and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic four-manifolds, 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 four-manifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic four-manifolds, which leads
Lagrangian matching invariants for fibred four-manifolds: I
, 2008
"... In a pair of papers, we construct invariants for smooth four-manifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the Donaldson-Smith 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 four-manifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the Donaldson-Smith invariant for Lefschetz fibrations. The ‘Lagrangian matching invariants ’ are designed
FOUR-MANIFOLDS WITHOUT EINSTEIN METRICS
- MATHEMATICAL RESEARCH LETTERS 3, 133–147 (1996)
, 1996
"... It is shown that there are infinitely many compact simply connected smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2χ>3|τ|. The examples in question arise as non-minimal 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 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2χ>3|τ|. The examples in question arise as non-minimal 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 three-manifold invariants developed in [21] and [20]. Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional 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 four-manifolds which bound a given threemanifold
Singular connection and Riemann theta function
, 1997
"... We prove the Chern-Weil 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 Chern-Weil 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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252
