Alexander module of a space ($H_1({\tilde{X}}_{\alpha };\mathbb{Z})$)

Given an integral epimorphism $\alpha :{\pi }_1(X)\to \mathbb{Z}$, the Alexander module is the first integral homology of the corresponding infinite cyclic cover associated to an epimorphism, with $t$ acting by its chosen deck transformation. Its order is the greatest common divisor of maximal presentation matrix minors when it is finitely presented and torsion. For the meridional epimorphism of a knot exterior, this is the Alexander module of a knot.