Floer homotopy theory and degenerate Lagrangian intersections

TL;DR

This paper establishes new lower bounds on Lagrangian intersection points using Floer homotopy theory and algebraic tools like Steenrod squares and quantum cap products, connecting Floer and Morse theories.

Contribution

It introduces a novel approach to bounding Lagrangian intersections by combining Floer homotopy types with Steenrod squares and Conley index computations.

Findings

Lower bounds on intersection points via Steenrod squares

Lower bounds using quantum cap product

Computation of Floer homotopy type in terms of Morse theory

Abstract

We give a lower bound on the number of intersection points of a Lagrangian pair via Steenrod squares on Lagrangian Floer cohomology induced from a Floer homotopy type. The main technical input is a computation of the associated graded of the action-filtration of the Floer homotopy type in terms of Morse homotopy theory (precisely, Conley index theory). We also prove a lower bound using the quantum cap product on Lagrangian Floer cohomology.