Morse moves in flow categories

TL;DR

This paper develops Morse move analogues for framed flow categories, enabling simplification of these categories without altering their associated stable homotopy types, thus aiding computations in Morse and Floer theories.

Contribution

It introduces Morse move analogues for framed flow categories, facilitating their simplification while preserving the associated stable homotopy types.

Findings

Analogues of Whitney trick and handle cancellation are established for framed flow categories.

These moves enable easier computation of stable homotopy types.

Application to Khovanov and Khovanov-Rozansky cohomologies demonstrates practical utility.

Abstract

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent index and handle cancellation which removes two critical points connected by a single flowline. A framed flow category is a way of encoding flow data such as that which may arise from the flowlines of a Morse function or of a Floer functional. The Cohen-Jones-Segal construction associates a stable homotopy type to a framed flow category whose cohomology is designed to recover the corresponding Morse or Floer cohomology. We obtain analogues of the Whitney trick and of handle cancellation for framed flow categories: in this new setting these are moves that can be performed to simplify a framed flow category without changing the associated stable homotopy type. These moves often enable one to compute by hand the stable homotopy type associated to a framed flow category. We apply this in the setting of the Lipshitz-Sarkar stable homotopy type (corresponding to Khovanov cohomology) and the stable homotopy type of a matched diagram due to the authors (corresponding to sl_n Khovanov-Rozansky cohomology).