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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.
Keywords: stable homotopy, knots, Khovanov, Lipshitz–Sarkar, Floer.
Lobb, Andrew 1 ; Orson, Patrick 2 ; Schütz, Dirk 1
@article{10_2140_agt_2018_18_2821,
author = {Lobb, Andrew and Orson, Patrick and Sch\"utz, Dirk},
title = {Framed cobordism and flow category moves},
journal = {Algebraic and Geometric Topology},
pages = {2821--2858},
publisher = {mathdoc},
volume = {18},
number = {5},
year = {2018},
doi = {10.2140/agt.2018.18.2821},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.2821/}
}
TY - JOUR AU - Lobb, Andrew AU - Orson, Patrick AU - Schütz, Dirk TI - Framed cobordism and flow category moves JO - Algebraic and Geometric Topology PY - 2018 SP - 2821 EP - 2858 VL - 18 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.2821/ DO - 10.2140/agt.2018.18.2821 ID - 10_2140_agt_2018_18_2821 ER -
%0 Journal Article %A Lobb, Andrew %A Orson, Patrick %A Schütz, Dirk %T Framed cobordism and flow category moves %J Algebraic and Geometric Topology %D 2018 %P 2821-2858 %V 18 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.2821/ %R 10.2140/agt.2018.18.2821 %F 10_2140_agt_2018_18_2821
Lobb, Andrew; Orson, Patrick; Schütz, Dirk. Framed cobordism and flow category moves. Algebraic and Geometric Topology, Tome 18 (2018) no. 5, pp. 2821-2858. doi: 10.2140/agt.2018.18.2821
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