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Framed cobordism and flow category moves. (English) Zbl 1418.57023

Flow categories are categories which encode the type of data arising in the flowlines of a Morse function, or the moduli spaces of a Floer functional. A framed flow category has a geometric realisation with a well defined stable homotopy type. This paper studies several operations on flow categories, analogous to operations in Morse theory, and their effect on geometric realisations.
Introduced by R. L. Cohen et al. [Prog. Math. 133, 297–325 (1995; Zbl 0843.58019)], a framed flow category \((\mathcal{C}, \iota, \varphi)\) roughly consists of finitely many \(\mathbb{Z}\)-graded objects, and morphism spaces \(\mathcal{M}(x,y)\) between objects \(x,y\) which are \((|x|-|y|-1)\)-dimensional manifolds with corners. The morphism spaces fit together analogously to flowlines of Morse functions, or moduli spaces of a Floer functional. They are also equipped with immersions \(\iota_{x,y}\) into Euclidean spaces with corners, together with framings \(\varphi_{x,y}\), which behave in an appropriately coherent fashion.
The geometric realisation of \((\mathcal{C},\iota,\varphi)\) is a cell complex whose construction involves several auxiliary choices, but different choices lead to complexes related by suspension, and the construction yields a spectrum, which can be regarded as a stable homotopy type.
The first main result of the present paper (Section 3) considers an operation on framed flow categories analogous to handle slides in Morse theory. Given a framed flow category \((\mathcal{C}, \iota, \varphi)\), a category \((\mathcal{C}_S, \iota_S, \varphi_S)\) is obtained by introducing two new objects \(e,f\), analogous to cancelling handles, together with appropriate morphism spaces, immersions, and framings; and then cancelling \(f\) against a different object, using the handle cancellation procedure developed by D. Jones et al. [Indiana Univ. Math. J. 66, No. 5, 1603–1657 (2017; Zbl 1394.57012)]. This procedure parallels a method of handle sliding by introducing cancelling handles. The authors prove (Theorem 3.1) that the geometric realisations of \(\mathcal{C}\) and \(\mathcal{C}_S\) have the same stable homotopy type.
The second main result (Section 4) concerns an operation on framed flow categories analogous to the Whitney trick; the authors call it an “extended Whitney trick”, by comparison with the Whitney trick previously applied to framed flow categories by Jones-Lobb-Schütz [loc. cit.]. Given a framed flow category \((\mathcal{C}, \iota, \varphi)\) and a single morphism space \(\mathcal{M}(x,y) = M\), the authors define a new framed flow category \((\mathcal{C}_W, \iota_W, \varphi_W)\), where \(M\) is replaced by another manifold \(M'\) via a cobordism \(W\) between \(M\) and \(M'\); other morphism spaces, immersions, and framings are modified appropriately. The authors prove (Theorem 4.1) that the geometric realisations of \(\mathcal{C}\) and \(\mathcal{C}_W\) have the same stable homotopy type.
The authors (in Section 5) also make a sample calculation, simplifying a given framed flow category associated to the disjoint union of three trefoils.
Finally, the authors consider which framed flow categories can be obtained from others by use of handle cancellations, the extended Whitney trick, isotopies of the framing, and changes of auxiliary data. They show (Theorem 6.2) that, starting from any framed flow category and applying such moves, they can arrive at a framed flow category whose based chain complex is in primary Smith normal form; moreover the numbers of points in 0-dimensional morphism spaces give the differential of the chain complex. The authors conjecture that any two framed flow categories with realisations of the same stable homotopy type are related by such moves.

MSC:

57R58 Floer homology
37D15 Morse-Smale systems
55P42 Stable homotopy theory, spectra
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

[1] 10.1016/B978-044481779-2/50002-X
[2] 10.1016/0040-9383(91)90020-5 · Zbl 0735.55002
[3] 10.1098/rspa.1950.0098 · Zbl 0041.10201
[4] ; Cohen, Geometric aspects of infinite integrable systems. Sūrikaisekikenkyūsho Kōkyūroku, 883, 68, (1994)
[5] 10.1007/BF02052956 · Zbl 0153.53801
[6] 10.1512/iumj.2017.66.6136 · Zbl 1394.57012
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