Field theories, stable homotopy theory, and Khovanov homology (Q2800409)

The paper under review was motivated by the desire to refine the TQFT of \textit{M. Khovanov} [Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)] to the setting of stable homotopy theory. The approach by the authors is to refine the TQFT to a sophisticated enough functor that the construction of \textit{A. Elmendorf} and \textit{M. Mandell} [Adv. Math. 205, No. 1, 163--228 (2006; Zbl 1117.19001)] can be applied to produce a module over the connective \(k\)-theory spectrum. The techniques used are very different from those of \textit{R. Lipshitz} and \textit{S. Sarkar} [J. Am. Math. Soc. 27, No. 4, 983-1042 (2014; Zbl 1345.57014)], who produced the original stable homotopy refinement of Khovanov homology using the framed flow category technology of \textit{R. L. Cohen} et al. [Prog. Math. 133, 297--325 (1995; Zbl 0843.58019)].NEWLINENEWLINEIn fact, the authors observe that their general theorem cannot be applied naively to the Khovanov TQFT, so instead they first pass to a spin-cobordism version of the Khovanov TQFT to make their functor refinement. Interestingly, the spectrum achieved at the end is shown to be independent of choices of spin structure. The explanation of this is that there is a ``geometric principle'' at work, as the authors put it. This principle is the Lipshitz-Sarkar ladybug matching, and indeed the paper under review provides an elegant reason to consider the ladybug matching to be the heart of stable homotopy refinements of Khovanov homology. The authors state that the global choice which must be made between left and right ladybug matchings in [Lipshitz and Sarkar, loc. cit.] corresponds to the fact that the embedded mapping class group of an unknotted 2-torus with two boundary components inside a cylinder on the 2-sphere is not trivial, but is the cyclic group of order two.NEWLINENEWLINESince the publication of the paper under review, and of the Lipshitz-Sarkar paper, their respective stable homotopy refinements of Khovanov homology were shown to be stably homotopy equivalent to one another, as modules over the connective \(k\)-theory spectrum, by \textit{T. Lawson, R. Lipshitz} and \textit{S. Sarkar} [``Khovanov homotopy type, Burnside category, and products'', Preprint, \url{arXiv:1505.00213}], which is a preprint at the time of writing this review.