Algebraic & Geometric Topology

Framed cobordism and flow category moves

Andrew Lobb, Patrick Orson, and Dirk Schütz

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Framed flow categories were introduced by Cohen, Jones and 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 and Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalising Khovanov cohomology.

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.

Article information

Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2821-2858.

Received: 10 May 2017
Revised: 23 January 2018
Accepted: 22 March 2018
First available in Project Euclid: 30 August 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37D15: Morse-Smale systems 55P42: Stable homotopy theory, spectra 57M27: Invariants of knots and 3-manifolds

stable homotopy knots Khovanov Lipshitz–Sarkar Floer.


Lobb, Andrew; Orson, Patrick; Schütz, Dirk. Framed cobordism and flow category moves. Algebr. Geom. Topol. 18 (2018), no. 5, 2821--2858. doi:10.2140/agt.2018.18.2821.

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