Inspired by Thurston’s covering conjecture, the author studies irreducible 3-manifolds with infinite fundamental group and without a finite covering with positive first Betti number. He finds analogies between the homology of the finite covers of such manifolds and class field theory. A representative theorem is the following:
Theorem 10.1. Let \(p\) be a prime such that \(\text{rank} (H_1(M, \mathbb{F}_p)) \geq 4\). Suppose \(M\) has no finite cover with positive first Betti number. Let \(M_{i+1}\), \(i\geq 1\) be the maximal abelian \(p\)-covering of \(M_i \) (the Hilbert class covering). Let \(r_i= \text{rank} (H_1 (M_i, \mathbb{F}_p))\). Then (1) \(r_{i+1} \geq {r^2_i-r_i \over 2}\), (2) Let \(\widetilde M_{i+1}\) be a maximal elementary abelian \(p\)-covering of \(\widetilde M_i\) \((\widetilde M_1=M_1)\). Then \(H_1 (\widetilde M_{i+1}, Z)_{(p)}\) has exponent \(\geq p^{\widetilde r_i -1}\). In particular \(\widetilde r_i\) has super-exponential growth and the class tower is infinite.