Floer homotopy theory for monotone Lagrangians

TL;DR

This paper develops a new Floer homotopy framework for monotone Lagrangians, overcoming high-dimensional curvature issues, and introduces invariants with Steenrod algebra actions and spectral sequences that reveal new topological restrictions.

Contribution

It introduces N-truncated, R-oriented flow categories for monotone Lagrangians, enabling Floer invariants with Steenrod algebra actions and spectral sequences, extending the scope of Floer theory.

Findings

Constructed Floer invariants with Steenrod algebra actions.

Established spectral sequences for these invariants.

Applied to real projective spaces to find new topological restrictions.

Abstract

We circumvent one of the roadblocks in associating Floer homotopy types to monotone Lagrangians, namely the curvature phenomena occurring in high dimensions. Given $N \ge 3$ and $R$ a connective $\mathbb E_1$-ring spectrum, there is a notion of an $N$-truncated, $R$-oriented flow category, to which we associate a module prospectrum over the Postnikov truncation $\tau_{\le N - 3}R$. This endows ordinary Floer cohomology with an action of the Steenrod algebra over $\tau_{\le N-3}R$, and also induces certain generalized cohomology theories. We give sufficient conditions for a closed embedded monotone Lagrangian to admit such well-defined invariants for $N = N_\mu$ the minimal Maslov number, and $R = MU$ complex bordism. Finally, we formulate Oh-Pozniak type spectral sequences for these invariants, and show that in the case of $\mathbb{RP}^n \subset \mathbb{CP}^n$ they provide further restrictions on the topology of clean intersections with a Hamiltonian isotopy, not detected by ordinary Floer (co)homology.