On a question of Serre

Alexander Rahm (1)

1 Department of mathematics

Consider the Borel-Serre compactification [1] of the quotient of hyperbolic 3-space H by a finite index subgroup Γ in a Bianchi group, and in particular the following question which Serre poses on page 514 of the quoted article. Consider the map α induced on homology when attaching the boundary ∂(Γ \H) into the Borel-Serre compactification Γ \H of the orbit space Γ \H. Question 1. How can one determine the kernel of α (in degree 1) ? Serre uses a global topological argument and obtains the rank of the kernel of α in H1 ∂(Γ \H) . But he keeps asking what submodule precisely this kernel is. With a local topological study, we can decompose the kernel of α into its parts associated to each cusp, both in first and second degree homology.

Domaines: K-théorie et homologie [math.KT]

Alexander Rahm. On a question of Serre. 2012. ⟨hal-00670217v1⟩