VFC package

An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations for invariants in symplectic topology arising from “counting” pseudo-holomorphic curves. We introduce the notion of an implicit atlas on a moduli space, which is (roughly) a convenient system of local finite-dimensional reductions. We present a general intrinsic strategy for constructing a canonical implicit atlas on any moduli space of pseudo-holomorphic curves. The main technical step in applying this strategy in any particular setting is to prove appropriate gluing theorems. We require only topological gluing theorems, that is, smoothness of the transition maps between gluing charts need not be addressed. Our approach to virtual fundamental cycles is algebraic rather than geometric (in particular, we do not use perturbation). Sheaf-theoretic tools play an important role in setting up our functorial algebraic “VFC package”. We illustrate the methods we introduce by giving definitions of Gromov-Witten invariants and Hamiltonian Floer homology over ℚ for general symplectic manifolds. Our framework generalizes to the S 1 -equivariant setting, and we use S 1 -localization to calculate Hamiltonian Floer homology. The Arnold conjecture (as treated by Floer, by Hofer and Salamon, by Ono, by Liu and Tian, by Ruan, and by Fukaya and Ono) is a well-known corollary of this calculation.

References in zbMATH (referenced in 14 articles )

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  1. Ganatra, Sheel; Pardon, John; Shende, Vivek: Covariantly functorial wrapped Floer theory on Liouville sectors (2020)
  2. Abbondandolo, Alberto; Schlenk, Felix: Floer homologies, with applications (2019)
  3. Alves, Marcelo R. R.; Colin, Vincent; Honda, Ko: Topological entropy for Reeb vector fields in dimension three via open book decompositions (2019)
  4. Ionel, Eleny-Nicoleta; Parker, Thomas H.: Relating VFCs on thin compactifications (2019)
  5. Pardon, John: Contact homology and virtual fundamental cycles (2019)
  6. Ginzburg, Viktor L.; Gürel, Başak Z.; Macarini, Leonardo: Multiplicity of closed Reeb orbits on prequantization bundles (2018)
  7. Ionel, Eleny-Nicoleta; Parker, Thomas: The Gopakumar-Vafa formula for symplectic manifolds (2018)
  8. Lipshitz, Robert; Ozsvath, Peter S.; Thurston, Dylan P.: Bordered Heegaard Floer homology (2018)
  9. McDuff, Dusa; Wehrheim, Katrin: The fundamental class of smooth Kuranishi atlases with trivial isotropy (2018)
  10. Stevenson, Bret: A quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer’s metric (2018)
  11. Charest, François; Woodward, Chris: Floer trajectories and stabilizing divisors (2017)
  12. Ginzburg, Viktor L.; Gürel, Başak Z.: Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds (2016)
  13. Pardon, John: An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves (2016)
  14. Nelson, Jo: Automatic transversality in contact homology. I: Regularity (2015)