Khovanov Homotopy Type, Periodic Links and Localizations

Given an m-periodic link $L \subset S^3$ , we show that the Khovanov spectrum $\mathcal{X}_{L}$ constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of $\mathcal{X}_{L}$ to the equivariant Khovanov homology of $L$ constructed by the second author. The action of Steenrod algebra on the cohomology of $\mathcal{X}_{L}$ gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.