Alexander Modules and Iwasawa Modules

In this chapter, we shall introduce the differential module for a group homomorphism and show the Crowell exact sequence associated to a short exact sequence of groups. Applying these constructions to the Abelianization map of a link group, we obtain the Alexander module of a link and the exact sequence relating the Alexander module with the link module. The argument is purely group-theoretical and can be applied to pro-finite (pro-l) groups in a parallel manner to obtain the complete differential module and the complete Crowell exact sequence. Applying these constructions to a homomorphism from a Galois group with restricted ramification, we obtain the complete Alexander module for a set of primes and the exact sequence relating the complete Alexander module with a Galois (Iwasawa) module.