Volume 23, issue 1 (2019)

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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

We calculate πK(S) [1 2], the homotopy groups of K(S) away from 2, in terms of the homotopy groups of K(), the homotopy groups of P1 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 P1 and the homotopy groups of S.

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algebraic $K$–theory, sphere spectrum, Whitehead spectrum, Waldhausen $A$–theory, stable pseudoisotopy theory, cyclotomic trace
Mathematical Subject Classification 2010
Primary: 19D10, 55Q10
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
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