This is an afterthought on [this MO question](https://mathoverflow.net/questions/45376/z-48-and-moonshine-beyond-the-monster), and also on Gannon's book mentioned there, about $K\_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the third stable homotopy group of spheres, $\pi ^S\_3=\mathbb{Z}/24$. Does anyone know about such a connection?



$K_3(\mathbb{Z})$ and $\pi ^S_3$
We have $\pi\_3(\mathbb{S}) \cong \mathbb{Z}/24\{ \nu\}$ and $\pi\_3K(\mathbb{Z}) \cong \mathbb{Z}/48\{ \lambda \}$. As Achim suggested, the unit map $\mathbb{S} \to K(\mathbb{Z})$ induces on $\pi\_3$ the injection sending $\nu$ to $2\lambda$. 


See the first paragraph of Section 2 of '[Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere](https://www.math.uni-bielefeld.de/documenta/vol-13/22.pdf)' by Ausoni, Dundas, and Rognes. 


