Floer homotopy theory and degenerate Lagrangian intersections

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.

💡 Research Summary

The paper addresses the classical problem of estimating the minimal number of intersection points of a pair of Lagrangian submanifolds ( (L_0,L_1) ) inside a symplectic manifold ((M,\omega)). Building on the foundations of Floer theory, the author introduces a new lower bound that works even when the intersections are degenerate. The key novelty is the use of a Floer homotopy type—a stable homotopy theoretic object associated to the Lagrangian pair—together with Steenrod square operations on its cohomology.

The work is organized around three sets of assumptions. Assumption 1 (closed or Liouville (M), relatively exact Lagrangians, Hamiltonian isotopy) recovers classical results such as Arnold’s conjecture, Hofer–Floer’s cup‑length bound, and the Hirschi‑Porcelli (R)-cup‑length estimate. Assumption 2 (closed or exact Liouville (M), exact Lagrangians, vanishing annular symplectic area) allows the definition of a quantum cap product on Lagrangian Floer homology, leading to a “quantum cap‑length’’ bound (Theorem 1.7). The most technical framework is Assumption 3, which requires a framed brane structure (\Lambda) on the pair; under this hypothesis the Floer homotopy type (F_{J,\Lambda}(L_0,L_1)) exists (Cohen‑Jones‑Segal/Abouzaid‑Blumberg).

The Floer homotopy type is filtered by action values (\kappa_j). Theorem 1.13 describes the associated graded pieces: for each action level the spectrum fits into a cofibration sequence \