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Spanier-Whitehead duality in the $ K(2)$-local category at $ p=2$


Author: Irina Bobkova
Journal: Proc. Amer. Math. Soc. 148 (2020), 5421-5436
MSC (2010): Primary 55P25, 55P42, 55T25
DOI: https://doi.org/10.1090/proc/15078
Published electronically: September 4, 2020
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Abstract: The fixed point spectra of Morava $ E$-theory $ E_n$ under the action of finite subgroups of the Morava stabilizer group $ \mathbb{G}_n$, and their $ K(n)$-local Spanier-Whitehead duals can be used to approximate the $ K(n)$-local sphere in certain cases. For any finite subgroup $ F$ of $ \mathbb{G}_2$ at $ p=2$ we prove that the $ K(2)$-local Spanier-Whitehead dual of the spectrum $ E_2^{hF}$ is $ \Sigma ^{44}E_2^{hF}$. These results are analogous to the known results at height 2 and $ p=3$. The main computational tool we use is the topological duality resolution spectral sequence for the spectrum $ E_2^{h\mathbb{S}_2^1}$ at $ p=2$.


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Additional Information

Irina Bobkova
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
MR Author ID: 1124377
ORCID: 0000-0001-8775-2691
Email: ibobkova@math.tamu.edu

DOI: https://doi.org/10.1090/proc/15078
Received by editor(s): June 24, 2018
Received by editor(s) in revised form: December 24, 2018, July 2, 2019, and February 24, 2020
Published electronically: September 4, 2020
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, CA, during the Spring 2019 semester, and Grant No. DMS-1638352 and DMS-2005627
Communicated by: Mark Behrens
Article copyright: © Copyright 2020 American Mathematical Society