On a question of Serre

Details On a question of Serre On a question of Serre

2012-02-15

arxiv.org/abs/1202.3219v1

Mathematics

K-Theory and Homology

Scientific paper

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

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