We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

We construct an orientable ribbon surface $F \subset B^4$ which is universal in the following sense: any orientable 4-manifold $M \cong B^4 \cup \text{1-handles} \cup \text{2-handles}$ can be represented as a cover of $B^4$ branched over $F$.

We construct some series of polyhedral schemata which represent orientable closed connected 3-manifolds. We show that these manifolds have spines corresponding to certain balanced presentations of their fundamental groups. Then we study some covering properties of such manifolds and prove that many of them are cyclic branched coverings of lens spaces. Our theorems contain a number of published results from various sources as particular cases.

AMS 2000 Mathematics subject classification: Primary 57M12; 57M50; 57M60

Let A be a finite abelian group and M be a branched cover of an homology 3-sphere, branched over a link L, with covering group A. We show that H1(M; Z[1/|A|]) is determined as a Z[1/|A|][A]-module by the Alexander ideals of L and certain ideal class invariants.