The Khovanov homology of infinite braids

Islambouli, Gabriel; Willis, Michael C.

doi:10.48550/arxiv.1610.04582

Cited by 1 publication

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“…In a different direction, the Khovanov homotopy type admits a number of extensions. Lobb-Orson-Schütz and, independently, Willis proved that the Khovanov homotopy type stabilizes under adding twists, and used this to extended it to a colored Khovanov stable homotopy type [53,83]; further stabilization results were proved by Willis [83] and Islambouli-Willis [28]. Jones-Lobb-Schütz proposed a homotopical refinement of the sl n Khovanov-Rozansky homology for a large class of knots [30] and there is also work in progress in this direction of Hu-Kriz-Somberg [27].…”

Section: Properties and Applicationsmentioning

confidence: 98%

Spatial Refinements and Khovanov Homology

Lipshitz

Sarkar

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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We review the construction and context of a stable homotopy refinement of Khovanov homology. IntroductionIn the late 1920s, in his study of critical points and geodesics [61][62][63], Morse introduced what is now called Morse theory-using functions for which the second derivative test does not fail (Morse functions) to decompose manifolds into simpler pieces. The finite-dimensional case was further developed by many authors (see [9] for a survey of the history), and an infinite-dimensional analogue introduced by Palais-Smale [65,66,80]. In both cases, a Morse function f on M leads to a chain complex C * (f ) generated by the critical points of f . This chain complex satisfies the fundamental theorem of Morse homology: its homology H * (f ) is isomorphic to the singular homology of M . This is both a feature and a drawback: it means that one can use information about the topology of M to deduce the existence of critical points of f , but also implies that C * (f ) does not see the smooth topology of M . (See [59,60] for an elegant account of the subject's foundations and some of its applications.)In the 1980s, Floer introduced several new examples of infinite-dimensional, Morse-like theories [21][22][23]. Unlike Palais-Smale's Morse theory, in which the descending manifolds of critical points are finite-dimensional, in Floer's setting both ascending and descending manifolds are infinite-dimensional. Also unlike Palais-Smale's setting, Floer's homology groups are not isomorphic to singular homology of the ambient space (though the singular homology acts on them). Indeed, most Floer (co)homology theories seem to have no intrinsic cup product operation, and so are unlikely to be the homology of any natural space.In the 1990s, Cohen-Jones-Segal [14] proposed that although Floer homology is not the homology of a space, it could be the homology of some associated 2010 Mathematics subject classification. 57M25,55P42

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