Quantitative estimates for fractional Sobolev mappings in rational homotopy groups

Abstract

Let $\mathcal{N} \subset \mathbb{R}^M$ be a smooth simply connected compact manifold 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 group element ${\rm deg}([f]) \in \mathbb{R}$ in terms of its fractional Sobolev-norm or H\"older norm. That is, we show that for all $\beta \in (\beta_0({\rm deg}),1]$, \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{W^{\beta,\frac{N}{\beta}}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}{\beta}}, \] and \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{C^{\beta}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}{\beta}}. \] Here $C({\rm deg}) > 0$, $L({\rm deg}) \in \mathbb{N}$, $\beta_0({\rm deg}) \in (0,1)$ are computable from the rational homotopy group represented by ${\rm deg}$. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov's integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivi\`ere.