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?



Detection of stable homotopy by K-theory spectra
$K(\mathbb Z)$ 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](https://publications.ias.edu/node/404) Pierre Deligne, Le déterminant de la cohomologie. (Picard groupoids, section 4)


[A relevant survey](http://link.springer.com/chapter/10.1007/978-1-4613-9526-3_7): Stephen Mitchell, On the Lichtenbaum-Quillen Conjectures from a Stable Homotopy-Theoretic Viewpoint.


The place to start is [Quillen](http://www.mathunion.org/ICM-search/answer.cgi?first=quillen), especially Cohomology of Groups, ICM 1970.


