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Framed cobordism and flow category moves

Lobb, Andrew; Orson, Patrick; Schuetz, Dirk

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Patrick Orson


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.


Lobb, A., Orson, P., & Schuetz, D. (2018). Framed cobordism and flow category moves. Algebraic & geometric topology, 18, 2821-2858.

Journal Article Type Article
Acceptance Date Mar 22, 2018
Online Publication Date Aug 22, 2018
Publication Date Aug 22, 2018
Deposit Date Apr 24, 2018
Publicly Available Date Aug 29, 2018
Journal Algebraic and Geometric Topology
Print ISSN 1472-2747
Electronic ISSN 1472-2739
Publisher Mathematical Sciences Publishers (MSP)
Peer Reviewed Peer Reviewed
Volume 18
Pages 2821-2858


Published Journal Article (489 Kb)

Copyright Statement
First published in Geometry & Topology in volume 18 on 22 August 2018 published by Mathematical Sciences Publishers. © 2018 Mathematical Sciences Publishers. All rights reserved.

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