Khovanov homotopy type, Burnside category and products

Lawson, Tyler¹ ; Lipshitz, Robert² ; Sarkar, Sucharit³

¹ Department of Mathematics, University of Minnesota, Minneapolis, MN, United States
² Department of Mathematics, University of Oregon, Eugene, OR, United States
³ Department of Mathematics, University of California at Los Angeles, Los Angeles, CA, United States

Geometry & topology, Tome 24 (2020) no. 2, pp. 623-745.

Résumé

We give a new construction of a Khovanov stable homotopy type, or spectrum. We show that this construction gives a space stably homotopy equivalent to the Khovanov spectra constructed by Lipshitz and Sarkar  (J. Amer. Math. Soc. 27 (2014) 983–1042) and Hu, Kriz and Kriz (Topology Proc. 48 (2016) 327–360) and, as a corollary, that those two constructions give equivalent spectra. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several of Lipshitz and Sarkar’s conjectures. Finally, combining these results with Lipshitz and Sarkar’s computations (J. Topol. 7 (2014) 817–848) and refined $s$–invariant (Duke Math. J. 163 (2014) 923–952), we obtain new results about the slice genera of certain knots.

Détail

DOI : 10.2140/gt.2020.24.623

Classification : 55P42, 57M25
Keywords: Khovanov homotopy, Khovanov homology, flow categories

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