Stable homotopy refinement of quantum annular homology. (8th April 2021)

• Record Type: Journal Article
• Authors: Akhmechet, Rostislav; Krushkal, Vyacheslav; Willis, Michael
• Is Part Of: Compositio mathematica. Volume 157:Number 4(2021)
• Page Start: 710
• Page End: 769
• Publication Date: 2021-04-08
• Subjects: quantum annular homology, -- annular links, -- equivariant stable homotopy type, -- equivariant Burnside category; 57K18, -- 55P42, -- 55P91; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X20007721
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.