Quantitative estimates for fractional Sobolev mappings in rational homotopy groups

TL;DR

This paper provides quantitative estimates linking fractional Sobolev and Hölder norms of maps from spheres to manifolds with their rational homotopy group elements, extending classical degree theory to a broader homotopical context.

Contribution

It introduces computable bounds for rational homotopy group elements using fractional Sobolev and Hölder norms, generalizing previous degree estimates to rational homotopy groups.

Findings

Establishes bounds for rational homotopy degrees via Sobolev norms.

Provides bounds using Hölder norms for maps into manifolds.

Extends classical degree estimates to 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.