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

Blumberg, Andrew¹ ; Mandell, Michael²

¹ Department of Mathematics, The University of Texas, Austin, TX, United States
² Department of Mathematics, Indiana University, Bloomington, IN, United States

Geometry & topology, Tome 23 (2019) no. 1, pp. 101-134.

Résumé

We calculate ${\pi }_{\ast }K\left(\mathbb{S}\right)\otimes \mathbb{Z}\left[\frac{1}{2}\right]$, the homotopy groups of $K\left(\mathbb{S}\right)$ away from $2$, in terms of the homotopy groups of $K\left(\mathbb{Z}\right)$, the homotopy groups of $ℂP_{-1}^{\infty }$ and the homotopy groups of $\mathbb{S}$. 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 $ℂP_{-1}^{\infty }$ and the homotopy groups of $\mathbb{S}$.

Détail

DOI : 10.2140/gt.2019.23.101

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

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