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The Novikov conjecture for linear groups

Erik Guentner, Nigel Higson, Shmuel Weinberger (2005)

Publications Mathématiques de l'IHÉS

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C*-algebra theory.

The Roquette category of finite p -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

Let p be a prime number. This paper introduces the Roquette category ℛ p of finite p -groups, which is an additive tensor category containing all finite p -groups among its objects. In ℛ p , every finite p -group P admits a canonical direct summand ∂ P , called the edge of P . Moreover P splits uniquely as a direct sum of edges of Roquette p -groups, and the tensor structure of ℛ p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from ℛ p to...

The signature package on Witt spaces

Pierre Albin, Éric Leichtnam, Rafe Mazzeo, Paolo Piazza (2012)

Annales scientifiques de l'École Normale Supérieure

In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the ‘depth’ of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index—the analytic signature of  X —is well-defined....

The tangent complex to the Bloch-Suslin complex

Jean-Louis Cathelineau (2007)

Bulletin de la Société Mathématique de France

Motivated by a renewed interest for the “additive dilogarithm” appeared recently, the purpose of this paper is to complete calculations on the tangent complex to the Bloch-Suslin complex, initiated a long time ago and which were motivated at the time by scissors congruence of polyedra and homology of SL 2 . The tangent complex to the trilogarithmic complex of Goncharov is also considered.

The universal functorial Lefschetz invariant

Wolfgang Lück (1999)

Fundamenta Mathematicae

We introduce the universal functorial Lefschetz invariant for endomorphisms of finite CW-complexes in terms of Grothendieck groups of endomorphisms of finitely generated free modules. It encompasses invariants like Lefschetz number, its generalization to the Lefschetz invariant, Nielsen number and L 2 -torsion of mapping tori. We examine its behaviour under fibrations.

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