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13
Floer homology and knot complements
, 2003
"... 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. Usi ..."
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Cited by 238 (7 self)
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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.
HOLOMORPHIC DISKS AND THREEMANIFOLD INVARIANTS: PROPERTIES AND APPLICATIONS
, 2001
"... ... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a ..."
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Cited by 201 (31 self)
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... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. 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.
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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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 ” 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. In addition to these new applications, the techniques also provide alternate proofs of Donaldson’s diagonalizability theorem and the Thom conjecture for CP 2. 1.
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 grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces. 1.
On the Floer homology of plumbed threemanifolds
 Geom. Topol
"... Abstract. We calculate HF + for threemanifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres. These calculations can be used to determine also the Floer homology of othe ..."
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Cited by 93 (9 self)
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Abstract. We calculate HF + for threemanifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres. These calculations can be used to determine also the Floer homology of other threemanifolds, including the product of a circle with a genus two surface. 1.
Heegaard Floer homology and alternating knots
, 2002
"... In [23] we introduced a knot invariant for a nullhomologous knot K in an oriented threemanifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the threesphere. We give a combinato ..."
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Cited by 85 (17 self)
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In [23] we introduced a knot invariant for a nullhomologous knot K in an oriented threemanifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the threesphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (compare [24]). Applications include new restrictions on the Alexander polynomial of alternating knots.
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 to new proofs of the indecomposability theorem for symplectic fourmanifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of fourmanifolds along a certain class of threemanifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such threemanifolds.
Fukaya categories and deformations
 Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 351–360, Higher Ed
, 2002
"... Soon after their first appearance [7], Fukaya categories were brought to the attention of a wider audience through the homological mirror conjecture [14]. Since then Fukaya and his collaborators have undertaken the vast project of laying down the foundations, and as a result a fully general definiti ..."
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Cited by 35 (5 self)
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Soon after their first appearance [7], Fukaya categories were brought to the attention of a wider audience through the homological mirror conjecture [14]. Since then Fukaya and his collaborators have undertaken the vast project of laying down the foundations, and as a result a fully general definition is available [9, 6]. The task that symplectic geometers are now facing is to make these categories into an effective tool, which in particular means developing more ways of doing computations in and with them. For concreteness, the discussion here is limited to projective varieties which are CalabiYau (most of it could be carried out in much greater generality, in particular the integrability assumption on the complex structure plays no real role). The first step will be to remove a hyperplane section from the variety. This makes the symplectic form exact, which simplifies the pseudoholomorphic map theory considerably. Moreover, as far as Fukaya categories are concerned, the affine piece can be considered as a first approximation to the projective variety. This is a fairly obvious idea, even though its proper formulation requires some algebraic formalism of deformation theory. A basic question is the finitedimensionality of the relevant deformation spaces. As Conjecture 4 shows, we hope for a favourable answer in many cases. It remains to be seen whether this is really a viable strategy for understanding Fukaya categories in interesting examples. Lack of space and ignorance keeps us from trying to survey related developments, but we want to give at least a few indications. The idea of working relative to a divisor is very common in symplectic geometry; some papers whose viewpoint is close to ours are [12, 16, 3, 17]. There is also at least one entirely different approach to Fukaya categories, using Lagrangian fibrations and Morse theory [8, 15, 4]. Finally, the example of the twotorus has been studied extensively [18]. Acknowledgements. Obviously, the ideas outlined here owe greatly to Fukaya
Periodic Floer ProSpectra from the SeibergWitten equations
, 2002
"... We use finite dimensional approximation to construct from the SeibergWitten equations invariants of threemanifolds with b1 > 0 in the form of periodic prospectra. Their homology is the SeibergWitten Floer homology. Then we proceed to construct relative stable homotopy SeibergWitten invariant ..."
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Cited by 14 (2 self)
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We use finite dimensional approximation to construct from the SeibergWitten equations invariants of threemanifolds with b1 > 0 in the form of periodic prospectra. Their homology is the SeibergWitten Floer homology. Then we proceed to construct relative stable homotopy SeibergWitten invariants of fourmanifolds with boundary.