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Stable homotopy refinement of quantum annular homology

Published online by Cambridge University Press:  08 April 2021

Rostislav Akhmechet
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA ra5aq@virginia.edu
Vyacheslav Krushkal
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA krushkal@virginia.edu
Michael Willis
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095, USA msw188@ucla.edu
Corresponding

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.

Type
Research Article
Copyright
© The Author(s) 2021

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Footnotes

R.A. was supported by NSF RTG Grant DMS-1839968, V.K. was supported by NSF Grant DMS-1612159, and M.W. was supported by NSF FRG Grant DMS-1563615.

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