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## HOLOMORPHIC DISKS AND THREE-MANIFOLD INVARIANTS: PROPERTIES AND APPLICATIONS (2001)

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Citations: | 201 - 31 self |

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793 |
Homological algebra
- Cartan, Eilenberg
- 1956
(Show Context)
Citation Context ...em 10.3 says that HF ∞ (Y ) is a free Z[U, U −1 ]-module of rank one. The claim about HF ′ ∗ (Y ) then follows immediately from the universal coefficients theorem spectral sequence (see, for instance =-=[5]-=-).HOLOMORPHIC DISKS AND THREE-MANIFOLD INVARIANTS 83 11. Applications In this section, we prove the remaining results (Theorems 1.8 and 1.12) claimed in the introduction. 11.1. Complexity of three-ma... |

313 |
Dehn surgery on knots,
- Culler, Gordon, et al.
- 1987
(Show Context)
Citation Context ...1.9 should be compared with the result of Gordon and Luecke which states that no non-trivial surgery on a non-trivial knot in the three-sphere can give back the three-sphere, see [13], [14], see also =-=[6]-=-. 1.2. Second application: bounding the number of gradient trajectories. We give another application, to Morse theory over homology three-spheres. Consider the following question. Fix an integral homo... |

274 | Holomorphic disks and topological invariants for closed threemanifolds,
- Ozsvath, Szabo
- 2004
(Show Context)
Citation Context ...al genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology. 1. Introduction The present paper is a continuation of =-=[26]-=-, where we defined topological invariants for closed orientated, three-manifolds Y , equipped with a Spin c structure s ∈ Spin c (Y ). These invariants are a collection of Floer homology groups HF − (... |

228 |
A Stipsicz, 4{manifolds and Kirby calculus,
- Gompf
- 1999
(Show Context)
Citation Context ...e k = n + 6, and where the spiral on the right hand side of the picture meets the horizontal circle k −2 times. For a general discussion on constructing Heegaard decompositions from link diagrams see =-=[12]-=-. The picture is to be interpreted as follows. Attach a one-handle connecting the two little circles on the left, and another one handle connecting the two little circles on the right, to obtain a gen... |

189 |
J-holomorphic curves and quantum cohomology
- McDuff, Salamon
- 1994
(Show Context)
Citation Context .... ̂ HF(Yα,γ, sα,γ), HF + (Yα,γ, sα,γ), Proof. The fact that f ∞ α,β,γ is a chain map follows by counting ends of one-dimensional moduli spaces of holomorphic triangles (see Section 5 of [23]; compare =-=[20]-=-). Fix x ∈ Tα ∩ Tβ, y ∈ Tβ ∩ Tγ, w ∈ Tα ∩ Tγ, and consider moduli spaces of holomorphic triangles M(ψ) where ψ ∈ π2(x,y,w), sz(ψ) = s, and µ(ψ) = 1. The ends of this moduli space are50 PETER OZSVÁTH ... |

188 |
Lectures on the h-cobordism Theorem
- Milnor
- 1965
(Show Context)
Citation Context ...nal index zero critical points. To see this, extend f to a Morse function ˜ f, and first cancel off all new index zero critical points. This is a familiar argument from Morse theory (see for instance =-=[23]-=-): given another index zero critical point p ′ , there is some index one critical point a which admits a unique flow to p ′ (if there no such index one critical points, then p ′ would generate a Z in ... |

183 | Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary,
- Ozsvath, Szabo
- 2003
(Show Context)
Citation Context ... = 0, then for any Spin c structure s, HF ∞ (Y, s) ∼ = Z[U, U −1 ]. More generally, if the Betti number if b1(Y ) ≤ 2, HF ∞ is determined by H1(Y ; Z). This is no longer the case when b1(Y ) > 2 (see =-=[29]-=-). However, if we use totally twisted coefficients (i.e. twisting by Z[H 1 (Y ; Z)], thought of as a trivial Z[H 1 (Y ; Z)]-module), then HF ∞ (Y, s) is always determined by H1(Y ; Z) (Theorem 10.12).... |

179 |
Knots are determined by their complements.
- Gordon, Luecke
- 1989
(Show Context)
Citation Context ...n in [29]. Corollary 1.9 should be compared with the result of Gordon and Luecke which states that no non-trivial surgery on a non-trivial knot in the three-sphere can give back the three-sphere, see =-=[13]-=-, [14], see also [6]. 1.2. Second application: bounding the number of gradient trajectories. We give another application, to Morse theory over homology three-spheres. Consider the following question. ... |

172 |
A norm for the homology of 3-manifolds
- Thurston
- 1985
(Show Context)
Citation Context ...) > 0 in an oriented three-manifold with b1(Y ) > 0. If s is a Spinc structure for which HF+(Y, s) = 0, then∣∣〈c1(s), [Z]〉∣∣ ≤ 2g(Z)− 2. We can reformulate this result using Thurston’s seminorm; see =-=[35]-=-. If Z = ⋃k i=1 Zi is a closed surface with k connected components, let χ−(Z) = k∑ i=1 max(0,−χ(Zi)). The Thurston seminorm of a homology class ξ ∈ H2(Y ;Z) is then defined by Θ(ξ) = inf{χ−(Z) ∣∣Z ⊂ Y... |

150 |
The unregularized gradient flow of the symplectic action
- Floer
- 1988
(Show Context)
Citation Context ...recall the necessary constructions briefly here, and refer the interested reader to [23], where they are built up in detail. (These constructions are modifications of Floer’s original construction in =-=[8]-=-.) If one chooses a complex structure j over the two-manifold Σ, there is a naturally induced holomorphic structure on the g-fold symmetric power Sym g (Σ), denoted Sym g (j). We can interpret the loc... |

139 |
An instanton invariant for 3 manifolds,
- Floer
- 1988
(Show Context)
Citation Context ...ctural analogues in Seiberg-Witten theory, with some partial results already established. For instance, a surgery exact sequence (analogous to Theorem 1.7) was established for instanton homology, see =-=[9]-=-, [4]. Also, the algebraic structure of “Seiberg-Witten-Floer” homology for three-manifolds with positive first Betti number is still largely conjectural, but expected to match with the structure of H... |

124 | Holomorphic triangles and invariants for smooth four-manifolds,
- Ozsváth, Szabó
- 2006
(Show Context)
Citation Context ...s four-dimensional picture can already be found in the discussion on holomorphic triangles (c.f. Section 8 of [26] and Section 9 of the present paper). These four-manifold invariants are presented in =-=[30]-=-. Although the link with Seiberg-Witten theory was our primary motivation for finding the invariants, we emphasize that the invariants studied here require no gauge theory to define and calculate, onl... |

104 |
Coherent orientations for periodic orbit problems in symplectic geometry
- Floer, Hofer
- 1993
(Show Context)
Citation Context ...cient for all the applications given in the introduction, but we describe the signed refinement with future applications in mind. This signed refinement uses systems of coherent orientations (compare =-=[10]-=-). Fix x,y representing some fixed Spin c structure t over Y . We let B(φ) denote the space of maps from the strip into Sym g (Σ) representing φ which live in some suitable Sobolev space (see Section ... |

92 | Monopoles and contact structures.
- Kronheimer, Mrowka
- 1997
(Show Context)
Citation Context ...ss in X). 6.1.3. Spin c structures. There is a geometric interpretation of Spin c structures in four dimensions, analogous to Turaev’s interpretation of Spin c structures in three-dimensions, compare =-=[16]-=- and [11]. Let X be a four-manifold. We consider pairs (J, P), where P ⊂ X is a collection of finitely many points in X, and J is an almost-complex structure defined over X − P. We say that two pairs ... |

91 | Self dual instantons and holomorphic curves
- Dostoglou, Salamon
- 1994
(Show Context)
Citation Context ...theory. The close relationship between this theory and Seiberg-Witten theory should be apparent. For example, Conjecture 1.1 is closely related to the Atiyah-Floer conjecture (see [1], see also [31], =-=[7]-=-), a loose statement of which is the following. A Heegaard decomposition of an integral homology three-sphere Y = U0 ∪Σ U1 gives rise to a space M, the space of SU(2)-representations of π1(Σ) modulo c... |

77 |
Morse-Bott theory and equivariant cohomology, The Floer memorial volume,
- Austin, Braam
- 1995
(Show Context)
Citation Context ...oer” homology for three-manifolds with positive first Betti number is still largely conjectural, but expected to match with the structure of HF + in large degrees (compare [16], [21], [27]); see also =-=[3]-=- for some corresponding results in instanton homology. However, the geometric content of these homology theories, which gives rise to bounds on the number of gradient trajectories (Theorem 1.11 and Th... |

75 | Symmetric products of an algebraic curve. Topology, - Macdonald - 1962 |

69 |
The Seiberg-Witten equations and four-manifolds with boundary, preprint
- Froyshov
(Show Context)
Citation Context ...mology. We recall briefly the construction . Our presentation here follows the lectures of Kronheimer and Mrowka [16]. For more discussion, see [3] for the instanton Floer homology analogue, and also =-=[11]-=-, [21], [37]. Let Y be an oriented rational homology 3-sphere, and s ∈ Spin c (Y ). After fixing additional data (a Riemannian metric over Y and some perturbation) the Seiberg-Witten equations over Y ... |

61 |
Floer’s work on instanton homology, knots and surgery
- Braam, Donaldson
- 1995
(Show Context)
Citation Context ...l analogues in Seiberg-Witten theory, with some partial results already established. For instance, a surgery exact sequence (analogous to Theorem 1.7) was established for instanton homology, see [9], =-=[4]-=-. Also, the algebraic structure of “Seiberg-Witten-Floer” homology for three-manifolds with positive first Betti number is still largely conjectural, but expected to match with the structure of HF + i... |

61 |
Three-dimensional manifolds and their Heegaard diagrams,
- Singer
- 1933
(Show Context)
Citation Context ...oves, and the inverse to stabilization, Heegaard moves. It is a standard result that any two Heegaard diagram for a given three-manifold can be connected by a sequence of Heegaard moves (see [27] and =-=[28]-=-; see also Proposition 2.1 of [23]). Sometimes, it is convenient to fix an additional reference point z ∈ Σ − α1 − . . . − αg − β1 − . . . − βg. The collection data (Σ, {α1, ..., αg}, {β1, ..., βg}, z... |

59 |
Lectures on the h-cobordism theorem. Notes by L
- Milnor
- 1965
(Show Context)
Citation Context ...nal index-zero critical points. To see this, extend f to a Morse function f̃ , and first cancel off all new index-zero critical points. This is a familiar argument from Morse theory (see for instance =-=[24]-=-): given another index-zero critical point p′, there is some index-one critical point a which admits a unique flow to p′ (if there no such index-one critical points, then p′ would generate a Z in the ... |

58 | Circle–valued Morse Theory, Reidemeister torsion and Seiberg– Witten invariants of 3 manifolds
- Hutchings, Lee
(Show Context)
Citation Context ...ults connecting Seiberg-Witten theory over four-manifolds with the theory of pseudoholomorphic curves, see [32]. For discussions on S 1 -valued Morse theory and SeibergWitten invariants, see [33] and =-=[15]-=-. Gauge-theoretic invariants in three dimensions are closely related to smooth four-manifold topology: Floer’s instanton homology is linked to Donaldson invariants, Seiberg-WittenFloer homology should... |

58 | Seiberg-Witten monopoles on Seifert fibered spaces.
- Mrowka, Ozsvath, et al.
- 1997
(Show Context)
Citation Context ...L(p, q), s), HF SW SW from (L(p, q), s) and HFred (L(p, q), s) are isomorphic to T +, T − , and 0 respectively. Note that all the 3-manifolds Y = Ym,n from Section 3 are Seifert-fibered so we can use =-=[24]-=- to compute their Seiberg-Witten Floer homology. Proposition 4.4. Let Y = Ym,n denote the oriented 3-manifold obtained by +n surgery along the torus knot T2,2m+1. Suppose also that n > 6m. Then for ea... |

57 |
Torsion invariants of Spin c –structures on 3–manifolds
- Turaev
- 1997
(Show Context)
Citation Context ...case where b1(Y ) = 1, τ(s) is calculated in the “chamber” containing c1(s). For zero-surgery on a knot, there is a well-known formula for the Turaev torsion in terms of the Alexander polynomial, see =-=[35]-=-. With this, the above theorem has the following corollary (a more precise version of which is given in Proposition 10.14, where the signs are clarified): Corollary 1.3. Let Y0 be the three-manifold o... |

42 |
Taubes, SW ⇒ Gr: from the Seiberg-Witten equations to pseudoholomorphic curves
- H
- 1996
(Show Context)
Citation Context ...irect analogue in Seiberg-Witten theory; but it is interesting to compare it with Taubes’ results connecting Seiberg-Witten theory over four-manifolds with the theory of pseudoholomorphic curves, see =-=[32]-=-. For discussions on S 1 -valued Morse theory and SeibergWitten invariants, see [33] and [15]. Gauge-theoretic invariants in three dimensions are closely related to smooth four-manifold topology: Floe... |

38 | Torsion invariants of Spinc -structures on 3-manifolds,
- Turaev
- 1997
(Show Context)
Citation Context ...case where b1(Y ) = 1, τ(s) is calculated in the “chamber” containing c1(s). For zero-surgery on a knot, there is a well-known formula for the Turaev torsion in terms of the Alexander polynomial; see =-=[36]-=-. With this, the above theorem has the following corollary (a more precise version of which is given in Proposition 10.14, where the signs are clarified): Corollary 1.3. Let Y0 be the three-manifold o... |

36 |
Symmetric products of an algebraic curve
- Macdonald
- 1962
(Show Context)
Citation Context ...any fixed z ∈ Σ, the subset {z} × Sym g−1 (Σ) ⊂ Sym g (Σ) is a complex submanifold.HOLOMORPHIC DISKS AND THREE-MANIFOLD INVARIANTS 9 There is a natural identification π1(Sym g (Σ)) ∼ = H1(Σ; Z) (see =-=[18]-=-). Also, for g > 1, π2(Sym g (Σ)) ∼ = Z, which is generated by a sphere S whose intersection number with {z} × Sym g−1 (Σ) is +1. The pairing between the first Chern class of the tangent bundle TSym g... |

32 |
Floer homology for Seiberg-Witten monopoles, in preparation
- Kronheimer, Mrowka
(Show Context)
Citation Context ...ection 4 we compare them to invariants with corresponding “equivariant SeibergWitten-Floer homologies”HF SWto , HF SW from, and HF SW red ; for the three-manifolds studied in Section 3, compare [21], =-=[16]-=-. These calculations support the following conjecture: Conjecture 1.1. Let Y be an oriented rational homology three-sphere. Then for all Spinc structures s ∈ Spinc(Y ) there are isomorphisms2 HF SWto ... |

31 | Equivariant Seiberg-Witten Floer homology, preprint
- Marcolli, Wang
(Show Context)
Citation Context ...n Section 4 we compare them with invariants with corresponding “equivariant Seiberg-Witten-Floer homologies”HF SW to , HF SW from , and HF SW red for the three-manifolds studied in Section 3, compare =-=[21]-=-, [16]. These calculations support the following conjecture: PSO was supported by NSF grant number DMS 9971950 and a Sloan Research Fellowship. ZSz was supported by a Sloan Research Fellowship and a P... |

26 | Scalar curvature and the Thurston norm,
- Kronheimer, Mrowka
- 1997
(Show Context)
Citation Context ... including the Euler characteristic calculation, which has its natural analogue in Seiberg-Witten theory (see [22], [36]), and the adjunction inequalities, which exist in both worlds (compare [2] and =-=[17]-=-). Two additional results presented in this paper – the surgery exact sequence and the algebraic structure of the Floer homology groups which follow from the HF ∞ calculations – have analogues in Floe... |

25 | Higher type adjunction inequalities in Seiberg-Witten theory
- Ozsváth, Szabó
(Show Context)
Citation Context ...Spin c structures determined by the line-bundles E, K ⊗ E −1 respectively. In order to simplify the computation we will use a certain perturbation of the SeibergWitten equation. Using the notation of =-=[25]-=- this perturbation depends on a real parameter u, and corresponds to adding a two-form iu(∗dη) to the curvature equation, where η is the connection form for Y over the orbifold. Now holomorphic soluti... |

24 |
A combinatorial formulation for the Seiberg–Witten invariants of 3–manifolds
- Turaev
- 1998
(Show Context)
Citation Context ...ctions 3 and 4, the close connection is also illustrated by several of the theorems, including the Euler characteristic calculation, which has its natural analogue in Seiberg-Witten theory (see [22], =-=[36]-=-), and the adjunction inequalities, which exist in both worlds (compare [2] and [17]). Two additional results presented in this paper – the surgery exact sequence and the algebraic structure of the Fl... |

24 | Seiberg-Witten-Floer stable homotopy type of three-manifolds with b1
- Manolescu
(Show Context)
Citation Context ...mology theories contain more information than Turaev’s torsion. This can be seen, for instance, from their behaviour under connected sums, which 2 This manuscript was written before the appearance of =-=[19]-=- and [20]. In the second paper, Kronheimer and Manolescu propose alternate Seiberg-Witten constructions, and indeed give one which they conjecture to agree with our ̂ HF.HOLOMORPHIC DISKS AND THREE-M... |

19 |
Taubes: The geometry of the Seiberg-Witten invariants
- H
(Show Context)
Citation Context ...ubes’ results connecting Seiberg-Witten theory over four-manifolds with the theory of pseudoholomorphic curves, see [32]. For discussions on S 1 -valued Morse theory and SeibergWitten invariants, see =-=[33]-=- and [15]. Gauge-theoretic invariants in three dimensions are closely related to smooth four-manifold topology: Floer’s instanton homology is linked to Donaldson invariants, Seiberg-WittenFloer homolo... |

17 |
CH Taubes, SW=Milnor torsion
- Meng
- 1996
(Show Context)
Citation Context ... of Sections 3 and 4, the close connection is also illustrated by several of the theorems, including the Euler characteristic calculation, which has its natural analogue in Seiberg-Witten theory (see =-=[22]-=-, [36]), and the adjunction inequalities, which exist in both worlds (compare [2] and [17]). Two additional results presented in this paper – the surgery exact sequence and the algebraic structure of ... |

16 |
Lagrangian intersections, 3-manifolds with boundary, and the AtiyahFloer conjecture
- Salamon
- 1994
(Show Context)
Citation Context ...gauge theory. The close relationship between this theory and Seiberg-Witten theory should be apparent. For example, Conjecture 1.1 is closely related to the Atiyah-Floer conjecture (see [1], see also =-=[31]-=-, [7]), a loose statement of which is the following. A Heegaard decomposition of an integral homology three-sphere Y = U0 ∪Σ U1 gives rise to a space M, the space of SU(2)-representations of π1(Σ) mod... |

14 | Periodic Floer pro-spectra from the Seiberg-Witten equations
- Kronheimer, Manolescu
(Show Context)
Citation Context ...eories contain more information than Turaev’s torsion. This can be seen, for instance, from their behaviour under connected sums, which 2 This manuscript was written before the appearance of [19] and =-=[20]-=-. In the second paper, Kronheimer and Manolescu propose alternate Seiberg-Witten constructions, and indeed give one which they conjecture to agree with our ̂ HF.HOLOMORPHIC DISKS AND THREE-MANIFOLD I... |

13 |
Integer graded instanton homology groups for homology three-spheres, Topology 31
- Fintushel, Stern
- 1992
(Show Context)
Citation Context ...h ⊗ [y, j] be given by the Maslov index of φ. In view of this, we can think of the corresponding homologies as analogues of a construction of Fintushel and Stern, for Z graded instanton homology (see =-=[8]-=-). For any Z[H1 (Y ; Z)]-module M, we have homology groups defined by ( HF(Y, s; M) = H∗ CF(Y, s) ⊗Z[H1 (Y ;Z)] M ) (where HF can be any of HF ∞ , HF + , HF − , or ̂ HF). The homology groups from [26]... |

13 | Holomorphic disks and topological invariants for rational homology threespheres. math.SG/0101206
- Ozsváth, Szabó
- 2000
(Show Context)
Citation Context ...rsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology. 1. Introduction In =-=[23]-=-, we defined topological invariants for closed, oriented three-manifolds Y with b1(Y ) = 0. Starting with a Heegaard diagram for Y , with Riemann surface Σ and attaching circles {α1, ..., αg} and {β1,... |

11 |
Instanton homology and Dehn surgery.
- Floer
- 1995
(Show Context)
Citation Context ...ness is then proved using a filtration on the homology groups above, together with the homological-algebraic constructions used in establishing the surgery sequences for instanton Floer homology (see =-=[10]-=-, [4]). The proof occupies the rest of the present subsection. Lemma 9.2. There is a pointed Heegaard multi-diagram with the property that (Σ, α, β, γ, δ, z) (1) the Heegaard diagrams (Σ, α, β), (Σ, α... |

10 |
The Thurston norm and three-dimensional Seiberg-Witten theory,
- Auckly
- 1996
(Show Context)
Citation Context ...heorems, including the Euler characteristic calculation, which has its natural analogue in Seiberg-Witten theory (see [22], [36]), and the adjunction inequalities, which exist in both worlds (compare =-=[2]-=- and [17]). Two additional results presented in this paper – the surgery exact sequence and the algebraic structure of the Floer homology groups which follow from the HF ∞ calculations – have analogue... |

9 |
are determined by their complements
- Knots
- 1989
(Show Context)
Citation Context ... [30]. Corollary 1.9 should be compared with the result of Gordon and Luecke which states that no nontrivial surgery on a nontrivial knot in the three-sphere can give back the three-sphere; see [13], =-=[14]-=- and also [6]. 1.2. Second application: bounding the number of gradient trajectories. We give another application, to Morse theory over homology three-spheres. Consider the following question. Fix an ... |

7 | Seiberg-Witten monopoles on Seifert spaces - Mrowka, Ozsvath, et al. - 1997 |

5 | The unregularized gradient of the symplectic action - Floer - 1988 |

5 | The Seiberg-Witten equations and four-manifolds with boundary - Fr¿yshov - 1996 |

4 | The theta divisor and three{manifold invariants, preprint
- Ozsvath, Szabo
(Show Context)
Citation Context ...l sign depends on i, j and g. (It is straightforward to verify that this geometric interpretation is equivalent to the more algebraic definition of ∆i,j given in [35], see for instance Section 7 from =-=[28]-=-.) Choose i and j so that both α∗ i and β∗ j have non-zero image in H2 (Y ; R). When b1(Y ) > 1, Turaev’s torsion is characterized by the equation (1) τ(s) − τ(s + α ∗ i) − τ(s + β ∗ j ) + τ(s + α ∗ i... |

3 |
Dirac operators, holomorphic Euler characteristics, and lattice points, preprint
- Kronheimer, Mrowka, et al.
- 1997
(Show Context)
Citation Context ...ns C + (E) correspond to effective divisors with degE < deg(K) − u 2 deg(N) , 2 and anti-holomorphic solutions C− (E) correspond to effective divisors with degE < deg(K) 2 + u deg(N) . 2 According to =-=[18]-=- the expected dimension of the moduli space between the reducible solution θ and C ±(E) is computed by dimM(θ, C ± ( ∑ (E)) = 1 + 2 χ(E ⊗ N i ) ) , i∈I ± -2 -2 -2 -m-1 -2 -2 . . . -2 Figure 5.22 PETE... |

3 | A norm for the homology of 3-manifolds, volume 59 of Mem - Thurston - 1986 |

2 |
A norm for the homology of 3-manifolds, volume 59
- Thurston
- 1986
(Show Context)
Citation Context ... in an oriented three-manifold with b1(Y ) > 0. If s is a Spin c structure for which HF +(Y, s) ̸= 0, then ∣ 〈c1(s), [Z]〉 ∣ ≤ 2g(Z) − 2. We can reformulate this result using Thurston’s semi-norm, see =-=[34]-=-. If Z = ⋃ k i=1 Zi is a closed surface with k connected components, let k∑ χ−(Z) = max(0, −χ(Zi)). The Thurston semi-norm of a homology class ξ ∈ H2(Y ; Z) is then defined by i=1 Θ(ξ) = inf{χ−(Z) ∣ ∣... |

2 | Absolutely graded homologies and intersection forms for fourmanifolds with boundary. math/0110170 - Ozsvath, Szabo - 2001 |

2 | Variants of equivariant Seiberg-Witten Floer homology. math.GT/0211238
- Marcolli, Wang
(Show Context)
Citation Context ...e appearance of [19] and [20]. In the second paper, Kronheimer and Manolescu propose alternate Seiberg-Witten constructions, and indeed give one which they conjecture to agree with our ̂ HF, see also =-=[22]-=-.HOLOMORPHIC DISKS AND THREE-MANIFOLD INVARIANTS 3 is studied in Section 6. (Recall that if Y1 and Y2 are a pair of three-manifolds both with positive first Betti number, then the Turaev torsion of t... |

1 | On embedding circle-bundles in four-manifolds - Ozsváth, Szabó - 2000 |

1 |
curvature and the Thurston norm
- Scalar
- 1997
(Show Context)
Citation Context ... including the Euler characteristic calculation, which has its natural analogue in Seiberg-Witten theory (see [23], [37]), and the adjunction inequalities, which exist in both worlds (compare [2] and =-=[17]-=-). Two additional results presented in this paper — the surgery exact sequence and the algebraic structure of the Floer homology groups which follow from the HF∞ calculations — have analogues in Floer... |

1 |
of equivariant Seiberg-Witten Floer homology
- Variants
(Show Context)
Citation Context ...e appearance of [19] and [20]. In the second paper, Kronheimer and Manolescu propose alternate Seiberg-Witten constructions, and indeed give one which they conjecture to agree with our ĤF ; see also =-=[22]-=-. HOLOMORPHIC DISKS AND THREE-MANIFOLD INVARIANTS 1161 Then, for each i = 0, χ(HF+(Y0, s0 + iH)) = ± d∑ j=1 ja|i|+j , where s0 is the Spinc structure with trivial first Chern class, and H is a genera... |

1 | embedding circle-bundles in four-manifolds - On |