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?

Topology from the differential viewpoint. $\endgroup$ – Theo Johnson-Freyd Jan 12 '19 at 15:381more comment