The integral homology of $PSL_2$ of imaginary quadratic integers with non-trivial class group

TL;DR

This paper computes the integral homology of Bianchi groups associated with imaginary quadratic fields with non-trivial class groups using a cellular complex, extending previous results known only for trivial class groups.

Contribution

It introduces a cellular complex approach to determine the integral homology of Bianchi groups for fields with non-trivial class groups, covering new cases m=5,6,10,13,15.

Findings

Computed integral homology for m=5,6,10,13,15

Extended known homology results to non-trivial class groups

Validated the cellular complex method for these computations

Abstract

We show that a cellular complex described by Floege allows to determine the integral homology of the Bianchi groups $PSL_2(O_{-m})$, where $O_{-m}$ is the ring of integers of an imaginary quadratic number field $\rationals[\sqrt{-m}]$ for a square-free natural number $m$. We use this to compute in the cases m = 5, 6, 10, 13 and 15 with non-trivial class group the integral homology of $PSL_2(O_{-m})$, which before was known only in the cases m = 1, 2, 3, 7 and 11 with trivial class group.