Three-manifolds class field theory (Homology of coverings for a non-virtually Haken manifold)

TL;DR

This paper explores the homology of finite coverings of three-manifolds, linking their algebraic properties to number field analogies, and introduces new techniques for computing homology with implications for the Thurston conjecture.

Contribution

It introduces a novel spectral sequence for homology computation and connects the pro-p completion of the fundamental group to Poincaré duality, advancing the understanding of three-manifold structures.

Findings

Pro-p completion of non-Thurston manifolds is a Poincaré duality pro-p group.

Developed a new spectral sequence for homology calculations.

Applications to finite group cohomology rings.

Abstract

This is a first in a series of papers, devoted to the relation betwwen three-manifolds and number fields. The present paper studies first homology of finite coverings of a three-manifold with primary interest in the Thurston $b_1$ conjecture.The main result reads: if $M$ does not yield the Thurston conjecture, then the pro-p completion of its fundamental group is a Poincar\'e duality pro-p group. Conceptually, it means that we have a ``p-adic'' three-manifold. We develop several algebraic techniques, including a new powerful specral seguence, to actually compute homology of coverings, assumong only information on homology of $M$, a thing never done before.A number of applications to the structure of finite group cohomology rings is also given.