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

TL;DR

This paper computes the homotopy groups of the algebraic K-theory of the sphere spectrum away from 2, relating them to known groups of K(Z), complex projective space, and the sphere spectrum, extending prior work.

Contribution

It provides explicit calculations of these homotopy groups in terms of established spectra, advancing understanding of algebraic K-theory of the sphere spectrum.

Findings

Calculated $oldsymbol{ ext{pi}_*K(oldsymbol{ ext{S}})[1/2]}$ in terms of known groups.

Extended previous work by Waldhausen, Quillen, Borel, and Rognes.

Connected homotopy groups of $K(oldsymbol{ ext{S}})$ to those of $K(oldsymbol{ ext{Z}})$, ${oldsymbol{ ext{CP}}^oldsymbol{ ext{infty}}_{-1}}$, and $oldsymbol{ ext{S}}$.

Abstract

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