The integral homology of $PS{L}_2$ of imaginary quadratic integers with non-trivial class group

Abstract

We show that a cellular complex described by Floege allows to determine the integral homology of the Bianchi groups $PS{L}_2(O_{-m})$, 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 $PS{L}_2(O_{-m})$, which before was known only in the cases m = 1, 2, 3, 7 and 11 with trivial class group.

Cited by in corpus (11)

• Resolutions for unit groups of orders
• Homology and K-theory of the Bianchi groups
• On Level One Cuspidal Bianchi Modular Forms
• Higher torsion in the Abelianization of the full Bianchi groups
• On the integral homology of PSL4(Z) and other arithmetic groups
• A refined Bloch group and the third homology of SL_2 of a field
• On Ruan's Cohomological Crepant Resolution Conjecture for the complexified Bianchi orbifolds
• Equivariant K-homology of Bianchi groups with non-trivial class group
• Introductory computations in the cohomology of arithmetic groups
• On the cohomology of linear groups over imaginary quadratic fields
• An introduction to torsion subcomplex reduction