On twisted torsion of compact 3-manifolds

Let M be a 3-manifold with connected non-vacuos boundary which is not spherical. Assume that N is another 3-manifold with vacuous boundary and N∗ is the 3-manifold obtained by removing from N the interior of a 3-cell. In the present paper, we find a relationship between the multiplicative property of the twisted Reidemeister torsion and the connected sum operation on these manifolds in order to understand their topology and geometry.