Let $G\subset \mathrm{SU}(2)$ be a finite group. (These are famously classified through the McKay correspondence.) The Lie group framing of $\mathrm{SU}(2) = S^3$ descends to the quotient manifold $S^3 / G$, at least after getting some "left"s and "right"s in the correct places.


Every framed $k$-manifold determines a class in the $k$th stable homotopy group of spheres $\pi\_k(\text{Sphere})$. For example, $\mathrm{SU}(2)$ with its Lie group framing provides a generator of $\pi\_3(\text{Sphere}) = \mathbb{Z}/24$.


What are the values of the homotopy-group classes determined by the framed 3-manifolds $S^3/G$? How do these values relate to other Lie theoretic data like the rank or (dual) Coxeter number of the ADE Dynkin diagram?



What is the value of $[S^3/G] \in \pi_3(Sphere)$ for a finite subgroup $G \subset SU(2)$?
The answer can be found in Theorem 2.1 from a paper of José Seade and Brian Steer (*Complex singularities and the framed cobordism class of compact quotients of $3$-dimensional Lie groups by discrete subgroups*, Comment. Math. Helv. **65** (1990), no. 3, 349–374, available [here](http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002067641)): If $G$ is the cyclic group of order $r$, then $S^3/G$ represents $r$ times a generator of $\pi\_3^s=\mathbb{Z}/24\mathbb{Z}$, whereas if $G$ is the $\langle p,q,r\rangle$ triangle group, then $S^3/G$ represents $(p+q+r-1)$ times a generator of $\pi\_3^s$.


