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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Framed cobordism and flow category moves

Andrew Lobb, Patrick Orson and Dirk Schütz

Algebraic & Geometric Topology 18 (2018) 2821–2858
Abstract

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.

Keywords
stable homotopy, knots, Khovanov, Lipshitz–Sarkar, Floer.
Mathematical Subject Classification 2010
Primary: 37D15, 55P42, 57M27
References
Publication
Received: 10 May 2017
Revised: 23 January 2018
Accepted: 22 March 2018
Published: 22 August 2018
Authors
Andrew Lobb
Department of Mathematical Sciences
Durham University
Durham
United Kingdom
Patrick Orson
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Dirk Schütz
Department of Mathematical Sciences
Durham University
Durham
United Kingdom