Eisenstein parts of homology and cohomology groups of Bianchi 3-fold

Abstract

Let K = Q( Ô≠d) where d(> 0) is a square-free integer. Let OK be the ring of integers of K. Consider the hyperbolic 3-space H3 ( Upper half space), H3 := {(z, t) œ C ◊ R | t > 0}. We define the extended 3- dimensional upper half space to be H3 := H3 fi K fi {Œ}. We denote the full Bianchi group SL2(OK) by G and choose to be a subgroup of SL2(OK) of finite index with no elements of finite order. Let Y = \H3 be a hyperbolic 3-manifold. Consider the Baily-Borel-Satake compactification of Y, which is XBB = \H3, obtained by adding the set of cusps. The Borel-Serre compactification of Y, which is XBS obtained by adding a 2-torus to each cusp ˆXBS (except for K = Q(i) or K = Q( Ô≠3) for which we add spheres instead). The first result of this thesis is related to the Eisenstein cycle and the Eisenstein part of homology. We explicitly write down the Eisenstein cycles (or we say Eisenstein element) in the first homology groups of quotients of hyperbolic 3-space as linear combinations of Cremona symbols (a generalization of Manin symbols) for imaginary quadratic fields. These cycles generate the Eisenstein part of the homology groups. Using Poincaré duality, we can relate cohomology and homology. We also studied the Eisenstein part of the cohomology groups. The second result of this thesis is related to the Eisenstein and cuspidal parts of the cohomology groups. We have calculated the trace of the first and second Eisenstein cohomology groups and the Lefschetz number. As an application of J.Rohlfs’ result in §8.4.1, we find an asymptotic dimension formula (in the level aspect) for the cuspidal cohomology groups of congruence subgroups of the form 1(N) inside the full Bianchi groups.