\section{Orientability of Sobolev line fields}\label{sec:orientability_of_sobolev_line_fields}

\footnote{\textcite{q43-sobolev-liftings} solves a lot of it - but for $0<s<1$ but I have $s =1$. And the referenced work only looks at lifting $𝕊^1$-valued map to $ℝ$ (angle function) and the other referenced work is supposed to show sth about arbitrary coverings but only works with $ℝ → 𝕊^1$. Weird on-first-sight-wrong-citation.}

% \footnote{\textcite{q43-sobolev-liftings} solves a lot of it - but for $0<s<1$ but I have $s = 1$. And the referenced work only looks at lifting $𝕊^1$-valued map to $ℝ$ (angle function) and the other referenced work is supposed to show sth about arbitrary coverings but only works with $ℝ → 𝕊^1$. Weird on-first-sight-wrong-citation.}

Since algebraic topology is only concerned with continuous maps it does not give us tools to directly study the orientability of Sobolev line fields.

Instead we use approximation results that help us to reduce the question to continuous fields.