Abstract
Given an m-periodic link Lsubset 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.
Highlights
Khovanov homology [27] is a link invariant that assigns to any diagram D ⊂ R2 of a link L ⊂ S3 a bigraded cochain complex CKh∗,∗(D), whose homology groups, Communicated by Thomas Schick.Kh∗,∗(D), are a link invariant
Lipshitz and Sarkar [34] showed that the action of stable cohomology operations on the Khovanov homology might lead to substantially stronger link invariants
We show in Theorem 8.3 that equivariant Khovanov homology is isomorphic to the Borel equivariant cohomology of XKh
Summary
Khovanov homology [27] is a link invariant that assigns to any diagram D ⊂ R2 of a link L ⊂ S3 a bigraded cochain complex CKh∗,∗(D), whose homology groups, Communicated by Thomas Schick. (2) The equivariant annular stable homotopy type of XAqK,kh(Dm) is an invariant of the associated annular m-periodic link. (2) The equivariant stable homotopy type of XKqh(Dm) is an invariant of the associated m-periodic link. Lipshitz and Sarkar [34] showed that the action of stable cohomology operations on the Khovanov homology might lead to substantially stronger link invariants. As a corollary of our construction, we obtain Theorem 8.10, which gives a functorial way to determine the annular Khovanov homology of the quotient link from the equivariant (annular) Khovanov homology of a periodic link. By Theorem 1.3 we know that XKZhp (and XAZKph) are determined by XKh (and XAKh) of the corresponding quotient link This gives a passage from the Khovanov homology of a p-periodic link to the Khovanov homology of the quotient (with coefficients in Zp). A careful study of stable cohomology operations leads to a refinement of the periodicity criterion of [5], for which we refer the reader to a separate paper [39]
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