TITLE: Detection of stable homotopy by K-theory spectra
QUESTION [9 upvotes]: This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from the map $\mathbb{S}\to K(R)$? I have heard that $K_1(\mathbb{Z})$ is generated by $\eta$ but don't have any idea of even where to start. Is this something people have investigated at all?

REPLY [8 votes]: I would say that the historically correct place to start is Quillen's letter to Milnor on the image of $(\pi_i O \to \pi_i^s \to K_i\mathbb{Z})$, published in Springer LNM 551 (1976). There Quillen proved that most of the image of $J$ in $\pi_*^s$ is detected in $K_*\mathbb{Z}$, including the honorary classes $\mu_{8k+1}$ and $\eta\mu_{8k+1}$ in degrees $8k+1$ and $8k+2$ that are detected in $\pi_* KO$ but are not in the image from $\pi_iO$.
The cases not covered by Quillen were the multiples $\eta \alpha_{8k-1}$ and $\eta^2 \alpha_{8k-1}$ in degrees $8k$ and $8k+1$, where $\alpha_{8k-1}$ generates the image-of-$J$ in degree $8k-1$ for $k\ge1$.  Waldhausen showed that these classes map to zero in $K_*\mathbb{Z}$, in Corollary 3.6 of his paper "Algebraic $K$-theory of spaces, a manifold approach" CMS Conf. Proc. 2 (1982).
The cokernel-of-$J$ maps to zero in $K_*\mathbb{Z}$; see Mitchell's "The Morava $K$-theory of algebraic $K$-theory spectra" $K$-Theory (1990).
Later on, these results became direct consequences of the proven Lichtenbaum-Quillen conjecture, as formulated in terms of etale $K$-theory by Dwyer-Friedlander in "Étale K-theory and arithmetic" Bull. AMS (1982).  See also the discussion about $JK(\mathbb{Z})$ in Bokstedt's "The rational homotopy type of $\Omega Wh^{Diff}(*)$" in Springer LNM 1051 (1984).
For any other (discrete) ring $R$, the unit map $S \to K(R)$ factors through $K(\mathbb{Z})$, so nothing more can be detected that way.  For strict ring spectra $R$ a larger part of $\pi_*^s$ can be detected, up to all of $\pi_*^s$ in the case $R = S$, since $S$ splits off from $K(S) = A(*)$.  As an intermediary example, for $p\ge5$ and $R = \ell$, the $p$-local Adams summand in connective topological $K$-theory, $\pi_*( K(\ell) ;\mathbb{Z}/p )$ detects $\beta'_1 \in \pi_*( S; \mathbb{Z}/p )$, as Ausoni and I showed in "Algebraic $K$-theory of topological $K$-theory" Acta Math (2002).