Lagrangian matching invariants for fibred four-manifolds: I

By Timothy Perutz. 2006

90 citations recorded by the source corpus.

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 invariants for Lefschetz fibrations. The ‘Lagrangian matching invariants’ are designed to be comparable with the Seiberg‐ Witten invariants of the underlying four-manifold; formal properties and first computations support the conjecture that equality holds. They fit into a field theory which assigns Floer homology groups to three-manifolds fibred over S 1 . The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres, subject to Lagrangian boundary conditions. Part I—the present paper—is devoted to the symplectic geometry of these Lagrangians.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4fc333e3c7a5a79152ddab4981470b20eed65d76

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Citations in the corpus

Fukaya categories of symmetric products and bordered Heegaard-Floer homology (2010, background): “As to the closed α-curves, using Perutz’s construction they determine a Lagrangian correspondence Tα from Sym g(F ) to Symḡ(Σ) (or, equivalently, T̄α from Sym ḡ(Σ) to Symg(F )).”
Taubes’s proof of the Weinstein conjecture in dimension three (2009, methodology): “For some more recent work relating this story to Seiberg-Witten theory, see, e.g., [43, 52 , 70].”
Fukaya categories and bordered Heegaard-Floer homology (2010, methodology): “By the work of Lekili and Perutz [4], the quilted Floer homology of the sequence (L1, . . . , Lr) is independent of the choice of the Morse function f and isomorphic to ĤF (Y ).”
Quilted Floer cohomology (2009, methodology): “Perutz [26] proved the following analogue of the Gysin sequence.”
Existence of broken Lefschetz fibrations (2008, unknown): “G T ] 1 2 Fe b 20 08”
Floer cohomology and geometric composition of Lagrangian correspondences (2009, methodology): “Perutz [13, 14] proposed a construction of three and four-manifold invariants using Floer theory for Lagrangian correspondences in symmetric products, which generalize the tori in Heegard Floer homology [OS].”
Quiver algebras as Fukaya categories (2013, background): “This is a Morse–Bott Lefschetz degeneration of the generic smooth fibre P(1) P(1) , so the monodromy is a fibred Dehn twist in the vanishing cycle; see Perutz [30].”
Wrinkled fibrations on near-symplectic manifolds (2007, methodology): “This has been verified by Perutz, [12], in several cases, notably in the case of symplectic manifolds as mentioned above, and when the underlying manifold is of the form S1 ×M3 , for any M which is a Z–homology–(S1 × S2) and for connected sums.”