# An estimate of the Hopf degree of fractional Sobolev mappings

## TL;DR

This paper provides an estimate of the Hopf degree for fractional Sobolev maps from high-dimensional spheres to spheres, using integral formulas and commutator estimates, extending understanding of topological invariants in fractional Sobolev spaces.

## Contribution

It introduces a new estimate for the Hopf degree of fractional Sobolev maps, connecting topological degree with fractional Sobolev seminorms for a range of smoothness levels.

## Key findings

- Establishes an upper bound for the Hopf degree in terms of fractional Sobolev seminorms.
- Uses Whitehead integral formula and commutator estimates in the proof.
- Extends the analysis of topological invariants to fractional Sobolev spaces.

## Abstract

We estimate the Hopf degree for smooth maps $f$ from $\mathbb{S}^{4n-1}$ to $\mathbb{S}^{2n}$ in the fractional Sobolev space. Namely we show that for $s \in [1 - \frac{1}{4n}, 1]$ \[ \left |{\rm deg}_H(f)\right | \lesssim [f]_{W^{s,\frac{4n-1}{s}}}^{\frac{4n}{s}}. \] Our argument is based on the Whitehead integral formula and commutator estimates for Jacobian-type expressions.