## The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

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Andrew J. Blumberg and Michael A. Mandell
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## Abstract:

Let $p\in {\mathbb {Z}}$ be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum ${\mathbb {S}}$ admits an “eigensplitting” that generalizes known splittings on $K$-theory and $TC$. We identify the summands in the fiber as the covers of ${\mathbb {Z}}_{p}$-Anderson duals of summands in the $K(1)$-localized algebraic $K$-theory of ${\mathbb {Z}}$. Analogous results hold for the ring ${\mathbb {Z}}$ where we prove that the $K(1)$-localized fiber sequence is self-dual for ${\mathbb {Z}}_{p}$-Anderson duality, with the duality permuting the summands by $i\mapsto p-i$ (indexed mod $p-1$). We explain an intrinsic characterization of the summand we call $Z$ in the splitting $TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z$ in terms of units in the $p$-cyclotomic tower of ${\mathbb {Q}}_{p}$.## References

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## Additional Information

**Andrew J. Blumberg**- Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
- MR Author ID: 648837
- Email: blumberg@math.columbia.edu
**Michael A. Mandell**- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 601045
- ORCID: 0000-0001-8442-3876
- Email: mmandell@indiana.edu
- Received by editor(s): February 16, 2022
- Received by editor(s) in revised form: August 4, 2022, and September 12, 2022
- Published electronically: December 16, 2022
- Additional Notes: The first author was supported in part by NSF grants DMS-1812064, DMS-2104420

The second author was supported in part by NSF grants DMS-1811820, DMS-2104348 - © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 2853-2874 - MSC (2020): Primary 19D10, 19D55, 19F05
- DOI: https://doi.org/10.1090/tran/8822
- MathSciNet review: 4557883