The twisted Alexander polynomial for finite abelian covers over three manifolds with boundary (Q422122)

If \(\widehat K\) is the lift of \(K\) in a finite cyclic branched cover of \(S^3\) along \(K\), and \(\Delta\) is the classical Alexander polynomial, there is a well-known formula relating \(\Delta_{\widehat K}\) and \(\Delta_K\). In this paper, this formula is generalized to a formula for the twisted Alexander polynomial of finite abelian covers of \(3\)-manifolds whose boundary is a finite union of tori. The authors take the viewpoint of Reidemeister torsion to define the polynomial torsion \(\Delta_M^{\varphi\otimes\rho}\) of a compact, connected \(3\)-manifold \(M\), where \(\varphi:\pi_1(M)\to\mathbb Z^n\) is a surjective homomorphism and \(\rho:\pi_1(M)\to\text{Aut}(V)\) is a representation for some complex vector space \(V\). The authors then establish a formula relating the polynomial torsions of \(\widehat M\) and \(M\).