This is primarily a reference request. Does anyone know of any writing about algebraic Ktheory 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?

3$\begingroup$ Rognes showed that modulo finite 2groups $\mathbb{S}\to K(\mathbb{Z})$ is at least 4connected in his paper "$K_4(\mathbb{Z})$ is the trivial group". Maybe you can extract a starting point from there? $\endgroup$– Jason PolakJul 22, 2014 at 19:08

3$\begingroup$ You may also look at jstor.org/stable/1971055?__redirected#mobileBookmark for the Relation between $\pi_3^s$ and $K_3({\mathbb Z})$. $\endgroup$– ThiKuJul 22, 2014 at 20:27
2 Answers
$K(\mathbb Z)$ 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 QuillenLichtenbaum 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 fullfledged 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 LichtenbaumQuillen Conjectures from a Stable HomotopyTheoretic Viewpoint.
The place to start is Quillen, especially Cohomology of Groups, ICM 1970.

2$\begingroup$ Yes, the $K$theory of discrete rings is height 1. But the redshift conjecture says that $K$theory raises height. So $K(E_n)$ is supposed to be height $n+1$ (ie, the localization is supposed to be an isomorphism in high degree). Somehow all discrete rings count as height 0, even in positive characteristic. $\endgroup$ Jul 23, 2014 at 3:56

2$\begingroup$ This reminds a bit of Dustin Clausen's paper arxiv.org/abs/1110.5851. In particular the real Jhomomorphism can be interpreted as a map of spectra $K(\mathbb{R}) \to \text{Pic}(\text{Sp})$. The author then constructs a "padic" version of this. Theorem 0.1 of cited document talks abou the image in the E(1)local sphere. $\endgroup$ Jul 23, 2014 at 4:05

3$\begingroup$ $p$adic rings detect everything because they map to characteristic zero fields. I guess my discussion didn't really cover characteristic zero fields or transcendental characteristic $p$ fields, but they detect everything. This reduces to the case of algebraically closed fields. The culmination of Suslin's rigidity argument is the calculation of the $\ell$adic completion of the $K$theory of an algebraically closed field as the $\ell$adic completion of $KU$, with practically no input. Thence the image of $J_U$. $\endgroup$ Jul 23, 2014 at 4:07

1$\begingroup$ QL is a precise statement about how $K(R)$ is almost $KU$local. Etale $K$theory is $KU$local and QL says that the comparison map is an isomorphism about the dimension. BL adds more detail. (Etale $K$theory is $KU$local because it is the global sections of a sheaf whose stalks are $KU$.) $\endgroup$ Jul 29, 2014 at 15:41

1$\begingroup$ @მამუკა ჯიბლაძე, Mainly I meant "above the dimension" ! ! ! The word "about" probably snuck from something like "is an isomorphism in degrees above a threshold, which is about the dimension." The dimension is the étale dimension. Usually it's stated for fields, where it is the Galois dimension. For a function field, that is equal to the dimension of the original variety (plus the dimension of the ground field). For a general variety, the Zariski dimension gets added in. $\endgroup$ Aug 3, 2014 at 16:29
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 $\eta \alpha_{8k1}$ and $\eta^2 \alpha_{8k1}$ in degrees $8k$ and $8k+1$, where $\alpha_{8k1}$ generates the imageof$J$ in degree $8k1$ 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 cokernelof$J$ 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 LichtenbaumQuillen conjecture, as formulated in terms of etale $K$theory by DwyerFriedlander in "Étale Ktheory 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).

$\begingroup$ Thanks John! This is a really great answer. I hadn't realized all of this was known. $\endgroup$ Dec 3, 2016 at 1:36