  Given a half-harmonic map $u\in \dot
H^{\frac{1}{2},2}(\mathbb{R},\mathbb{S}^1)$ minimizing the fractional Dirichlet
energy under Dirichlet boundary conditions in $\mathbb{R}\setminus I$, we show
the existence of a second half-harmonic map, minimizing the Dirichlet energy in
a different homotopy class. This is a first step towards addressing a problem
raised by Brezis and is based on the study of the degree of fractional Sobolev
maps and a sharp estimate \`a la Brezis-Coron. We give examples showing that it
is in general not possible to minimize in every homotopy class and show a
contrast with the 2-dimensional case.
