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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.

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    $\begingroup$ The definition of the Hopf invariant using differential forms in explained in the book by Bott and Tu, Differential Forms in Algebraic Topology, page 228. $\endgroup$ – Piotr Hajlasz Mar 13 '18 at 2:00
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    $\begingroup$ I doubt it. Differential forms do not detect torsion phenomena in homotopy--they are real invariants. $\endgroup$ – John Klein Mar 13 '18 at 3:11
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    $\begingroup$ @DavidRoberts integration along gives a chain map from the DeRham complex to the singular cochains (in your case with coefficients in the complex numbers. This map will not detect torsion. However, I should have modified my original comment. If one is willing to work with twisted coefficients in a flat bundle, the there are invariants which detect torsion phenomena (in homology), for example, the analytic torsion invariant of Ray and Singer. $\endgroup$ – John Klein Mar 13 '18 at 12:08
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    $\begingroup$ Bundle n-gerbes with connection are very closely related to differential cohomology and detect all integral cohomology groups. Many differential geometric constructions, such as Chern classes, admit a refinement to bundle n-gerbes with connection. $\endgroup$ – Dmitri Pavlov Mar 13 '18 at 19:41
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    $\begingroup$ @DeaneYang In the case of homotopy groups of spheres the rational homotopy works only if the homotopy groups is infinite which is exactly the case of $\pi_n(S^n)$ and $\pi_{4n-1}(S^{2n})$, but that is covered by differential forms. Did you men Sullivan's theory? $\endgroup$ – Piotr Hajlasz Jan 21 at 0:42

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