The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory

Abstract

The purpose of this paper is to extend the construction of the PSS-type isomorphism between the Floer homology and the quantum homology of a monotone Lagrangian submanifold $L$ of a symplectic manifold $M$, provided that the minimal Maslov number of $L$ is at least two, to arbitrary coefficients. We provide a proof, again over arbitrary coefficients, that this isomorphism respects the natural algebraic structures on both sides, such as the quantum product and the quantum module action. This isomorphism serves as the technical foundation for the construction of Lagrangian spectral invariants in a joint paper with Remi Leclercq (arXiv:1505:07430). Our constructions work when the second Stiefel--Whitney class of $L$ vanishes on the image of the boundary homomorphism ${\pi }_3(M,L)\to {\pi }_2(L)$, a condition strictly weaker than being relatively Pin; in particular we do not require $L$ to be orientable. The constructions are done using canonical orientations, and require no further choices such as relative Pin-structures. Such structures do however play a significant role when endowing the various complexes and homologies with structures of modules over Novikov rings, and in calculations.

arXiv comment133 pages. Introduction and abstract rewritten to better reflect the state of the art. Comments welcome

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Cited by in corpus (9)

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• Higher rank local systems in Lagrangian Floer theory
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• Hamiltonian no-torsion
• Function theoretical applications of Lagrangian spectral invariants
• Applications of the theory of Floer to symmetric spaces
• Floer homology of fibrations I: Representing flow lines in Moore path spaces
• Lagrangian intersections and a conjecture of Arnol'd
• The fundamental group of a rigid Lagrangian cobordism