I am studying Turaev's work on torsions of manifolds, specifically the paper Euler Structures, Nonsingular Vector Fields, and Torsions of Reidemeister Type. (I cannot find an open-access version of that to link to but see the letter Torsion Invariants of Spin-c-Structures on 3-Manifolds, specifically Section 1.3 ).
He defines a smooth Euler structure on a closed, oriented manifold to be a homology class $[u]$ of a non-singular vector field $u$ on $M$ (say $\dim M =m \ge 2$).
Two non-singular vector fields $u$ and $v$ are said to be homologous if for some closed $m$-dimensional ball $D \subset M$ the restrictions of $u$ and $v$ to $M \setminus \text{int}(D)$ are homotopic in the class of non-singular vector fields.
My question is, Why is this an interesting homology relation for non-singular vector fields? In particular, why do we take out the ball $D$ before searching for a homotopy? What would go wrong if we tried to look for homotopies of $u$ and $v$ on all of $M$, instead of just $M\setminus D$?
Turaev cites V. G. Boltyanskii, Homotopy Classification of Vector Fields, but I haven't been able to find this paper (and suspect that it is in Russian, which I can't read). Can anyone recommend a reference for more on this homology relation on vector fields?