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Let $\mathcal{N} \subset \mathbb{R}^M$ be a smooth simply connected compactmanifold without boundary. A rational homotopy subgroup of$\pi_{N}(\mathcal{N})$ is represented by a homomorphism \[{\rm deg}:\pi_{N}(\mathcal{N}) \to \mathbb{R}.\] For maps $f: \mathbb{S}^N \to\mathcal{N}$ we give a quantitative estimate of its rational homotopy groupelement ${\rm deg}([f]) \in \mathbb{R}$ in terms of its fractional Sobolev-normor H\"older norm. That is, we show that for all $\beta \in (\beta_0({\rmdeg}),1]$, \[  |{\rm deg}([f])|\leq C({\rm deg})\,[f]_{W^{\beta,\frac{N}{\beta}}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}{\beta}},\] and \[  |{\rm deg}([f])|\leq C({\rm deg})\,[f]_{C^{\beta}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}{\beta}}. \] Here $C({\rmdeg}) > 0$, $L({\rm deg}) \in \mathbb{N}$, $\beta_0({\rm deg}) \in (0,1)$ arecomputable from the rational homotopy group represented by ${\rm deg}$. Thisextends earlier work by Van Schaftingen and the second author on the Hopfdegree to the Novikov's integral representation for rational homotopy groups asdeveloped by Sullivan, Novikov, Hardt and Rivi\`ere.

Quantitative estimates for fractional Sobolev mappings in rational homotopy groups