The integral homology of [formula omitted] of imaginary quadratic integers with nontrivial class group

Abstract

We show that a cellular complex defined by Flöge allows us 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 Q [ − m ] for a square-free natural number m . In the cases of nontrivial class group, we handle the difficulties arising from the cusps associated to the nontrivial ideal classes of O − m . We use this to compute the integral homology of PS L 2 ( O − m ) in the cases m = 5 , 6 , 10 , 13 and 15 , which previously was known only in the cases m = 1 , 2 , 3 , 7 and 11 with trivial class group.