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43
Holomorphic disks and threemanifold invariants: properties and applications
"... ̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. The pr ..."
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Cited by 120 (28 self)
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̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to threemanifold topology. 1.
Floer homology and knot complements
, 2003
"... Abstract. We use the OzsváthSzabó theory of Floer homology to define an invariant of knot complements in threemanifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the OzsváthSzabó Floer homology of large integral surgeries on the k ..."
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Cited by 116 (7 self)
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Abstract. We use the OzsváthSzabó theory of Floer homology to define an invariant of knot complements in threemanifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the OzsváthSzabó Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of perfect knots in S3 for which ĈF r has a particularly simple form. For these knots, formal properties of the OzsváthSzabó theory enable us to make a complete calculation of the Floer homology. It turns out that most small knots are perfect. 1.
The Symplectic Thom Conjecture
"... In this paper, we demonstrate a relation among SeibergWitten invariants which arises from embedded surfaces in fourmanifolds whose selfintersection number is negative. These relations, together with Taubes' basic theorems on the SeibergWitten invariants of symplectic manifolds, are then used to ..."
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Cited by 43 (8 self)
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In this paper, we demonstrate a relation among SeibergWitten invariants which arises from embedded surfaces in fourmanifolds whose selfintersection number is negative. These relations, together with Taubes' basic theorems on the SeibergWitten invariants of symplectic manifolds, are then used to prove the Symplectic Thom Conjecture: a symplectic surface in a symplectic fourmanifold is genusminimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative selfintersection in fourmanifolds. 1.
Monopoles and lens space surgeries
 ArXive:math.GT/0310164
, 2003
"... Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which sh ..."
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Cited by 34 (10 self)
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Abstract. Monopole Floer homology is used to prove that real projective threespace cannot be obtained from Dehn surgery on a nontrivial knot in the threesphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of threemanifolds which do not admit taut foliations. 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.
Generic metrics, irreducible rankone PU(2) monopoles, and transversality
 Comm. Anal. Geom
"... Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given fourmanifold: see Theorem 1.3. In [16] we proved t ..."
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Cited by 18 (7 self)
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Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given fourmanifold: see Theorem 1.3. In [16] we proved transversality using an
SeibergWitten invariants of 4manifolds with free circle actions
 Commun. Contemp. Math
, 2001
"... The main result of this paper describes a formula for the SeibergWitten invariant of a 4manifold X which admits a nontrivial free S 1action. A free circle action on X is classified by its orbit space, a 3manifold M, and its Euler class χ ∈ H 2 (M; Z). If χ = 0, then ..."
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Cited by 17 (4 self)
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The main result of this paper describes a formula for the SeibergWitten invariant of a 4manifold X which admits a nontrivial free S 1action. A free circle action on X is classified by its orbit space, a 3manifold M, and its Euler class χ ∈ H 2 (M; Z). If χ = 0, then
Seifert fibered contact threemanifolds via surgery
, 2003
"... Using contact surgery we define families of contact structures on certain Seifert fibered three–manifolds. We prove that all these contact structures are tight using Ozsváth–Szabó’s contact invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered thre ..."
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Cited by 15 (7 self)
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Using contact surgery we define families of contact structures on certain Seifert fibered three–manifolds. We prove that all these contact structures are tight using Ozsváth–Szabó’s contact invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three–manifold carrying at least n pairwise non– isomorphic tight, not semi–fillable contact structures.
PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality
 J. DIFFERENTIAL GEOM
, 1997
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