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

1996/02/14 by Reznikov, Alexander
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.dg-ga/9602006

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 b₁ conjecture.The main result reads: if M does not yield the Thurston conjecture, then the pro-p completion of its fundamental group is a Poincaré 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.

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