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Stable homotopy refinement of quantum annular homology

Part of: Homotopy theory

Published online by Cambridge University Press:  08 April 2021

Rostislav Akhmechet ,
Vyacheslav Krushkal  and
Michael Willis
Rostislav Akhmechet
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA22904-4137, [email protected]
Vyacheslav Krushkal
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA22904-4137, [email protected]
Michael Willis
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA90095, [email protected]
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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.

Keywords

quantum annular homologyannular linksequivariant stable homotopy typeequivariant Burnside category

MSC classification

Secondary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 4 , April 2021 , pp. 710 - 769
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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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