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

TL;DR

This paper extends the PSS isomorphism between Floer and quantum homology for monotone Lagrangian submanifolds to arbitrary coefficients, ensuring it respects algebraic structures without requiring orientability or Pin-structures.

Contribution

It provides a construction of the PSS isomorphism with canonical orientations for a broader class of Lagrangians, weakening previous topological restrictions and simplifying orientation choices.

Findings

The isomorphism respects quantum product and module actions.

Construction works under weaker topological conditions, not requiring orientability.

No additional structures like Pin-structures are needed for the core isomorphism.

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.