Framed cobordism and flow category moves

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. 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.37D15, 55P42, 57M27

A FAST ALGORITHM FOR CALCULATING S-INVARIANTS

“…Using Gaussian elimination, we can ensure that the coboundary matrix of F n has only entries 0 and 2. We then use handle slides, compare, for example, [10], to turn F n into a direct sum of Z/4Z-elementary cochain complexes. This argument is similar to the one used in the proof of [10,Theorem 6.2], where a chain complex is brought into Smith Normal Form.…”

Khovanov Homotopy Calculations using Flow Category Calculus

“…Moreover, it was shown in [LS14b] that Steenrod squares in Khovanov cohomology can be used to refine the Rasmussen sinvariant of the smooth concordance class of a link, providing another important reason to be interested in their computation. The main purpose of this paper is to describe a combinatorial algorithm for calculating Steenrod squares in the cohomology of the suspension spectrum associated to a general framed flow category based on the flow category simplifications developed in [JLS15b,LOS16]. In particular this is a new method computing the Steenrod squares in Khovanov cohomology.…”

“…In Sections 4 and 5 we turn to the algorithm. The algorithm makes heavy use of the flow category moves in [JLS15b] and [LOS16], or rather their 1-flow category restrictions. The moves -called handle slides, handle cancellation, and Whitney trick -for modifying a framed flow category, are based on ideas from Morse theory.…”

“…The moves -called handle slides, handle cancellation, and Whitney trick -for modifying a framed flow category, are based on ideas from Morse theory. The moves do not change the stable homotopy type associated to the flow category, but we show in [LOS16,§6] that they can be used to simplify the flow category itself in an algorithmic way. To take advantage of this idea for our eventual Steenrod square algorithm, we describe explicitly in Section 4 how our flow category moves restrict to framed 1-flow categories.…”

“…Note that adding an object c with |c| = k + 2 as in Remark 3.18 establishes exactly this Adem relation.4. Morse moves in 1-flow categoriesIn[JLS15b] and[LOS16] the authors, together with Dan Jones, defined a series of flow category moves -called handle slides, handle cancellation, and Whitney trick -for modifying and algorithmically simplifying a framed flow category. Section 4 is dedicated to an explicit description of the restrictions of these moves to 1-flow categories, with a mind to their application in Section 5 as part of the new algorithm for computing Steenrod squares.4.1.…”