Two results on Eisenstein part of homology and cohomology groups of Bianchi modular groups

We explicitly write down the {\it Eisenstein cycles} in the first homology groups of quotients of the hyperbolic three spaces as linear combinations of Cremona symbols (a generalization of Manin symbols) for imaginary quadratic fields. They generate the Eisenstein part of the homology groups. We also study the Eisenstein part of the cohomology groups. As an application, we find an asymptotic dimension formula (level aspect) for the cuspidal cohomology groups of congruence subgroups of the form ${\text{Ga}}_1(N)$ inside the full {\it Bianchi groups}.