Lagrangian Floer theory over integers: spherically positive symplectic manifolds

TL;DR

This paper extends Lagrangian Floer theory to integer coefficients in spherically positive symplectic manifolds, using sheaf theory and stratification techniques, under certain assumptions.

Contribution

It demonstrates that Lagrangian Floer theory can be developed over 2 coefficients generally, and over 2 when Lagrangians are relatively spin, employing new technical tools.

Findings

Floer theory over 2 is possible under specific conditions.

Development over 2 is feasible for relatively spin Lagrangians.

Sheaf and stratification methods are key technical tools.

Abstract

In this paper we study the Lagrangian Floer theory over $\Z$ or $\Z_2$. Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in \cite{fooo-book} can be developed over $\Z_2$ coefficients, and over $\Z$ coefficients when Lagrangian submanifolds are relatively spin. The main technical tools used for the construction are the notion of the sheaf of groups, and stratification and compatibility of the normal cones applied to the Kuranishi structure of the moduli space of pseudo-holomorphic discs.