A polyfold proof of the Arnold conjecture

• Benjamin Filippenko ^[1] ; Katrin Wehrheim ^[2]
  1. [1] Stanford University

     Stanford University

     Estados Unidos

     
  2. [2] University of California System

     University of California System

     Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 1, 2022
• Idioma: inglés
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• Resumen

  • We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from CP1×M to C+×M and C−×M, as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.