Homology and K-theory of the Bianchi groups

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant $K$-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the $K$-theory of their reduced $C^*$-algebras in terms of isomorphic images of the computed $K$-homology. We further find an application to Chen/Ruan orbifold cohomology. % {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I +++ (2011).

A refinement of a conjecture of Quillen

We present some new results on the cohomology of a large scope of SL\_2-groups in degrees above the virtual cohomological dimension; yielding some partial positive results for the Quillen conjecture in rank one. We combine these results with the known partial positive results and the known types of counterexamples to the Quillen conjecture, in order to formulate a refined variant of the conjecture

The mod 2 cohomology rings of SL\_2 of the imaginary quadratic integers

We establish general dimension formulae for the second page of the equivariant spectral sequence of the action of the SL\_2 groups over imaginary quadratic integers on their associated symmetric space. On the way, we extend the torsion subcomplex reduction technique to cases where the kernel of the group action is nontrivial. Using the equivariant and Lyndon-Hochschild-Serre spectral sequences, we investigate the second page differentials and show how to obtain the mod 2 cohomology rings of our groups from this information.Comment: Version post-print corrigeant de petits erreurs concernant l'observation qui se r\'ef\`ere sur l'appendice, Journal of Pure and Applied Algebra, Elsevier, 2015, Accepted for publication on July 21, 201

Fault detection in combinational networks

This paper presents an algorithm for locating a failure in combinational logic networks, which is a problem of importance in the maintenance of computer systems. The procedure is based on the path sensitizing idea for fault detection. The networks considered are non-redundant, consisting of AND, OR, and NOT elements. The class of faults investigated is that which causes a connection to appear to be logically suck-at-one or stuck-at-zero, and only single failures are treated. It is shown that the failure is generally located to a specific fault group --Abstract, page 2

Complexifiable characteristic classes

We examine the topological characteristic cohomology classes of complexified vector bundles. In particular, all the classes coming from the real vector bundles underlying the complexification are determined.Comment: This is the final manuscript of the author, prepared before submission to the publisher. Journal of Homotopy and Related Structures (2013) Accepted for publicatio

Accessing the cohomology of discrete groups above their virtual cohomological dimension

We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL_2 over the ring of integers in an imaginary quadratic number field, and to their finite index subgroups. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups in the Bianchi groups have been determined in a thesis of Kr\"amer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups