On twisted torsion of compact $3$-manifolds

Abstract: 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 $N^{\ast}$ 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.