Lagrangian matching invariants for fibred four-manifolds: I

Perutz, Tim

doi:10.2140/gt.2007.11.759

Cited by 51 publications

(79 citation statements)

References 26 publications

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“…Quilted Floer homology was originally designed to construct symplectic versions of gauge theoretic invariants, in particular symplectic versions of Donaldson invariants, which we develop in later papers [42; 43], Seiberg-Witten invariants as in Perutz [25] and Lekili [16], and Khovanov invariants as in Rezazadegan [26]. Applications to symplectic topology are given for moduli spaces of flat bundles in [43], and to classification of Lagrangians in tori in Abouzaid-Smith [1].…”

Section: Applicationsmentioning

confidence: 99%

Quilted Floer cohomology

2010

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We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. This provides the foundation for the construction of a symplectic 2-category as well as for the definition of topological invariants via decomposition and representation in the symplectic category. Here we give some first direct symplectic applications: Calculations of Floer cohomology, displaceability of Lagrangian correspondences and transfer of displaceability under geometric composition. 53D40; 57R56

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Section: Applicationsmentioning

confidence: 99%

Quilted Floer cohomology

2010

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“…We note finally that the modules HF * (Y, π, r; w) fit into a field theory for Lefschetz fibrations over surfaces with boundary, which has been studied by M. Usher [16] and the author [12] (the latter extends the framework to a larger class of singular fibrations). This too is thought to be intimately related to Seiberg-Witten theory.…”

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confidence: 94%

Symplectic Fibrations and the Abelian Vortex Equations

Perutz

2007

Commun. Math. Phys.

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Abstract. The nth symmetric product of a Riemann surface carries a natural family of Kähler forms, arising from its interpretation as a moduli space of abelian vortices. We give a new proof of a formula of Manton-Nasir [10] for the cohomology classes of these forms. Further, we show how these ideas generalise to families of Riemann surfaces.These results help to clarify a conjecture of D. Salamon [13] on the relationship between Seiberg-Witten theory on 3-manifolds fibred over the circle and symplectic Floer homology.

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“…B for the map defining the relative symmetric product. Then picking p 2 C D @D , z j F 1 .C / is cohomologous to the pullback of j F 1 .p/ to S 1 F 1 .p/ as a result of the fact that they have Hamiltonian-equivalent monodromies, and so just as in the first paragraph of this proof we can interpolate between these two forms and so glue z j F 1 .C / to the pullback of z j According to [34,Lemma 3.15] (which for our present purposes replaces an erroneous lemma in [7] which we had referred to in earlier versions of this paper; we thank the referee for pointing out this issue and suggesting a way of resolving it), where Á 2 H 2 .EI ‫/ޚ‬ restricts to the singular fiber E 0 as the orientation class (so Á is a positive multiple of c C 2 PD.h/), any cohomology class in This positions us to state the result underlying the construction of the maps induced by fibered cobordisms. …”

Section: Maps From Cobordismsmentioning

confidence: 99%

Vortices and a TQFT for Lefschetz fibrations on 4–manifolds

Usher

2006

Algebr. Geom. Topol.

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Adapting a construction of D Salamon involving the U.1/ vortex equations, we explore the properties of a Floer theory for 3-manifolds that fiber over S 1 which exhibits several parallels with monopole Floer homology, and in all likelihood coincides with it. The theory fits into a restricted analogue of a TQFT in which the cobordisms are required to be equipped with Lefschetz fibrations, and has connections to the dynamics of surface symplectomorphisms.57R57; 57R56, 53D401 Background and summary of resultsFor some time it has been known that two of the most important invariants of smooth closed 4-manifolds, the Donaldson and Seiberg-Witten invariants, can each be expressed in terms of .3 C 1/-dimensional topological quantum field theories (see Donaldson [6], Marcolli and Wang [27], Kronheimer and Mrowka [19]). In such a "TQFT," to each oriented 3-manifold Y (perhaps equipped with additional data, such as a spin c -structure), one associates canonically a group V .Y / satisfying, among several other conditions, the property that a cobordism X from Y 1 to Y 2 functorially induces a homomorphismIf X is a smooth closed oriented 4-manifold, divided into two pieces as X D X 1 [ Y X 2 with b C .X i / > 0, one views X 1 as a cobordism from the empty set ¿ to Y and X 2 as a cobordism from ¿ to Y (ie, Y with its orientation reversed). One has a natural identification V . Y / Š V .Y / , and the 4-dimensional invariant I X is obtained by a natural calculation in V .Y / involving the images of the maps F X 1 and F X 2 ; I X is independent of the choice of splitting of X into the two pieces X 1 and X 2 .In the presence of a symplectic structure ! on the spin c 4-manifold .X; s/, the famous work of C Taubes collected in [45] shows that the Seiberg-Witten invariant S W X .s/ agrees with a "Gromov invariant" Gr .X ;!/ .˛s/ which counts pseudoholomorphic submanifolds of X representing a homology class˛s corresponding to s . Kronheimer and Mrowka's work [19] (see [20] for a summary) lays the full foundations for the TQFT underlying S W X .s/, in which the role of the group V .Y / in the above description is played by HM.Y; s; Á/, where s is a spin c structure and Á 2 H 2 .Y I ‫/ޒ‬ is the

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Lagrangian matching invariants for fibred four-manifolds: I

Cited by 51 publications

References 26 publications

Quilted Floer cohomology

Quilted Floer cohomology

Symplectic Fibrations and the Abelian Vortex Equations

Vortices and a TQFT for Lefschetz fibrations on 4–manifolds

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