Khovanov homotopy type, periodic links and localizations

TL;DR

This paper explores the action of symmetry groups on the Khovanov spectrum of periodic links, relating it to equivariant homology and providing a new proof of the localization formula.

Contribution

It introduces a homology group action on the Khovanov spectrum for periodic links and connects it to equivariant Khovanov homology, offering an alternative proof of the localization formula.

Findings

Homology group action on Khovanov spectrum for periodic links

Relation between Borel cohomology and equivariant Khovanov homology

New proof of the localization formula for Khovanov homology

Abstract

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 homology 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 Dwyer-Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.