TITLE: $K_3(\mathbb{Z})$ and $\pi ^S_3$
QUESTION [9 upvotes]: This is an afterthought on this MO question, 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?

REPLY [9 votes]: 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' by Ausoni, Dundas, and Rognes.