The only infinite homotopy groups of spheres are $\pi_n(\mathbb{S}^n)$ and $\pi_{4n-1}(\mathbb{S}^{2n})$. This is a well known result of Serre. In both cases the nontriviality of these groups can be detected using differential forms. Namely, using the degree theory in the case of $\pi_n(\mathbb{S}^n)$ and using the Hopf invariant in the case of $\pi_{4n-1}(\mathbb{S}^{2n})$. **Are there any known cases of finite non-trivial homotopy groups of spheres that can be detected with differential forms?**

There is a well known method of Pontryagin of computing homotopy groups of spheres which is based on methods of differential topology, as beautifully described in notes by Andy Putman. However, I am more interested in the use of differential forms.

I am interested in analytical methods since I am interested in Lipschitz homotopy groups. Classical homological methods do not distinguish between Lipschitz and continuous and so analytical methods are much more adequate.

Differential Forms in Algebraic Topology, page 228. $\endgroup$ – Piotr Hajlasz Mar 13 '18 at 2:00