Detection of stable homotopy by K-theory spectra

Question

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?

Answer 1

$K(\mathbb Z)$ detects the image of $J$ and Quillen's proof of the Adams conjecture about the image of $J$ is closely related to the $K$-theory of finite fields.

$S$ splits off of $K(S)$, if you are interested in such things.

The Quillen-Lichtenbaum conjecture says that $K(R)$ is pretty close to étale $K$-theory, which is $KU$-local. Indeed, Thomason showed that the $KU$-localization of $K(R)$ is étale $K$-theory; and Mitchell showed that the full chromatic localization gives nothing more. So it is unlikely to to detect elements which are not $KU$-local. Probably QL is enough to show that it does not happen.

So does $K(\mathbb Z)$ detect all $KU$-local elements, the image of $J$? Most are detected by $K(\mathbb F_\ell)$, and thus by the intermediate $K(\mathbb Z)$. $K(\mathbb F_\ell^\nu)$ has no $\ell$-torsion, but it detects almost all $p$-torsion. For odd $p$ and appropriate choice of $\ell$, the $p$-part of $K(\mathbb F_\ell)$ is the $p$-part of $J$. For wrong choice of $\ell$, it still detects them all, but just has more torsion. For $p=2$, it only detects the image of complex $J_U\colon U\to S$, not full-fledged real $J\colon O\to S$. I believe that the same is true of $K(\mathbb Z)$.

This argument did not depend much on the choice of ring. If the ring has a point of characteristic $\ell$, then it detects the $p$-torsion for all other $p$. If it is an $\mathbb F_p$-algebra, then its $K$-theory factors through $K(\mathbb F_p)$ and does not detect any $p$-torsion. This leaves only the case of $K(\mathbb Z/p^n)$, which detects some but not all $p$-torsion, in particular none with exponent exceeding $p^n$.

For $K_1$, see Pierre Deligne, Le déterminant de la cohomologie. (Picard groupoids, section 4)

A relevant survey: Stephen Mitchell, On the Lichtenbaum-Quillen Conjectures from a Stable Homotopy-Theoretic Viewpoint.

The place to start is Quillen, especially Cohomology of Groups, ICM 1970.

Answer 2

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).