Branched Spines of 3-Manifolds and Reidemeister Torsion of Euler Structures

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.