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Khovanov homotopy type, periodic links and localizations
Maciej Borodzik1, Wojciech Politarczyk1, Marithania Silvero2,3
1Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland.
Mathematische Annalen
|November 15, 2021
Summary
We reveal a group action on the Khovanov spectrum for periodic links, connecting Borel cohomology to equivariant Khovanov homology. This provides new insights into link homology and its algebraic structures.
Area of Science:
- Knot theory
- Algebraic topology
- Homological algebra
Background:
- Khovanov homology provides a powerful invariant for knots and links.
- Periodic links possess symmetries that can be exploited for deeper analysis.
- The Khovanov spectrum and equivariant Khovanov homology are advanced tools in the field.
Purpose of the Study:
- To investigate the structure of the Khovanov spectrum for m-periodic links.
- To establish a relationship between Borel cohomology and equivariant Khovanov homology.
- To explore the implications of Steenrod algebra actions and provide alternative proofs for existing results.
Main Methods:
- Construction of a group action on the Khovanov spectrum.
- Relating Borel cohomology to equivariant Khovanov homology.
- Application of the Dwyer-Wilkerson theorem.
Main Results:
- The Khovanov spectrum of an m-periodic link admits a group action.
- A connection is established between Borel cohomology and equivariant Khovanov homology.
- An alternative proof of the localization formula for Khovanov homology is provided.
- Khovanov homology of quotient links is expressed via equivariant Khovanov homology.
Conclusions:
- The study reveals novel algebraic structures associated with periodic links.
- The findings offer a new perspective on the localization formula in Khovanov homology.
- The work deepens the understanding of the interplay between different homology theories for links.
Keywords:
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