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

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

1996-02-14

arxiv.org/abs/dg-ga/9602006v1

Mathematics

Differential Geometry

amstex

Scientific paper

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.

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