We consider homotopy classes of non-singular vector
fields on three-manifolds with boundary and we define for these
classes torsion invariants of Reidemeister type. We show that
torsion is well-defined and equivariant under the action of the appropriate homology group using an elementary and self-contained
technique. Namely, we use the theory of branched standard spines
to express the difference between two homotopy classes as a combination of well-understood elementary catastrophes. As a special
case we are able to reproduce Turaev’s theory of Reidemeister torsion for Euler structures on closed manifolds of dimension three.
