Foundation of Floer homotopy theory I: Flow categories

TL;DR

This paper develops a new stable infinity category framework for Floer homotopy theory using flow categories and bordism theories, enabling direct construction and interpretation of Floer homotopy types and groups.

Contribution

It introduces a novel categorical approach to Floer homotopy theory, connecting flow categories with bordism theories and modeling spectra within this framework.

Findings

Constructed a stable infinity category with flow categories and bimodules.

Demonstrated Floer homotopy types as mapping spectra in this category.

Showed the category models spectra in the case of framed bordism.

Abstract

We construct a stable infinity category with objects flow categories and morphisms flow bimodules; our construction has many flavors, related to a choice of bordism theory, and we discuss in particular framed bordism and the bordism theory of complex oriented derived orbifolds. In this setup, the construction of homotopy types associated to Floer-theoretic data is immediate: the moduli spaces of solutions to Floer's equation assemble into a flow category with respect to the appropriate bordism theory, and the associated Floer homotopy types arise as suitable mapping spectra in this category. The definition of these mapping spectra is sufficiently explicit to allow a direct interpretation of the Floer homotopy groups as Floer bordism groups. In the setting of framed bordism, we show that the category we construct is a model for the category of spectra. We implement the construction of Floer homotopy types in this new formalism for the case of Hamiltonian Floer theory.