Khovanov homotopy type, periodic links and localizations (Q2049951)

A link \(L\subset S^3\) is called \(m\)-periodic, if there is an orientation preserving action of \(\mathbb{Z}/m\) on \(S^3\) such that \(L\) is an invariant subset and the fixed point set is an unknot disjoint from \(L\). Removing a tubular neighborhood of the fixed point set produces an annular link \(L\subset S^1\times D^2\) invariant under a fixed point free action. For an annular link its Khovanov complex admits a filtration whose graded complex is called the annular Khovanov complex. Work of \textit{R. Lipshitz} and \textit{S. Sarkar} [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)] produced a stable homotopy type for the Khovanov complex, and this construction can be lifted to the annular complex. Furthermore, it is known that the Khovanov complex of an \(m\)-periodic link admits an induced action of \(\mathbb{Z}/m\). One main result of this paper is a lifting of this action to the homotopy type, resulting in an equivariant stable homotopy type as an invariant of the \(m\)-periodic link. Furthermore, this also works for the annular homotopy type. The construction of this equivariant homotopy type is done via cubical framed flow categories, somewhat similar to the original construction of Lipshitz-Sarkar. However, getting invariance of the diagram requires more work than in the non-equivariant case. Another construction was given by \textit{M. Stoffregen} and \textit{M. Zhang} [``Localization in Khovanov homology'', Preprint, \url{arXiv:1810.04769}] using a different approach involving the Burnside category. It is left as an open question whether these constructions lead to the same invariant. The authors also obtain fixed point theorems in this situation, allowing them to relate the fixed point set in the Khovanov type of the \(m\)-periodic link with the annular type of the quotient link. This enables them to get Smith inequalities for the dimensions of the homology groups. More advanced techniques from homotopy theory are needed for these results.