The homotopy groups of the algebraic K–theory of the sphere spectrum

Blumberg, Andrew; Mandell, Michael

doi:10.2140/gt.2019.23.101

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The homotopy groups of the algebraic $K$–theory of the sphere spectrum

Andrew J Blumberg and Michael A Mandell

Geometry & Topology 23 (2019) 101–134

DOI: 10.2140/gt.2019.23.101

Abstract

We calculate $π_{*} K (S) \otimes ℤ [\frac{1}{2}]$ , the homotopy groups of $K (S)$ away from $2$ , in terms of the homotopy groups of $K (ℤ)$ , the homotopy groups of $ℂ P_{- 1}^{\infty}$ and the homotopy groups of $S$ . This builds on work of Waldhausen, who computed the rational homotopy groups (building on work of Quillen and Borel) and Rognes, who calculated the groups at odd regular primes in terms of the homotopy groups of $ℂ P_{- 1}^{\infty}$ and the homotopy groups of $S$ .

Keywords

algebraic $K$–theory, sphere spectrum, Whitehead spectrum, Waldhausen $A$–theory, stable pseudoisotopy theory, cyclotomic trace

Mathematical Subject Classification 2010

Primary: 19D10, 55Q10

References

Publication

Received: 4 April 2017

Revised: 1 May 2018

Accepted: 14 June 2018

Published: 5 March 2019

Proposed: Mark Behrens

Seconded: Jesper Grodal, Haynes R Miller

Authors

Andrew J Blumberg
Department of Mathematics The University of Texas Austin, TX United States

Michael A Mandell
Department of Mathematics Indiana University Bloomington, IN United States

Volume 23, issue 1 (2019)