Framed cobordism and flow category moves

TL;DR

This paper introduces moves for framed flow categories that preserve their associated stable homotopy type, aiming to simplify their study through combinatorial calculus, with applications to knot invariants like Khovanov cohomology.

Contribution

It defines a set of moves that alter framed flow categories without changing their stable homotopy type, facilitating their combinatorial analysis and simplification.

Findings

Moves preserve the stable homotopy type.

Examples demonstrate the moves' application to Khovanov categories.

Results show potential for simplifying framed flow categories.

Abstract

Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz-Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalizing Khovanov cohomology. In this paper we give moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse-Smale case without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories to a combinatorial calculus. We provide examples of calculations performed with these moves (related to the Khovanov framed flow category), and prove some general results about the simplification of framed flow categories via these moves.