Floer homotopy theory for monotone Lagrangians

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.

36 pages, 2 figures; added more acknowledgements and remarks