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A Künneth formula in topological homology and its applications to the simplicial cohomology of ℓ ¹ ( ℤ ₊ k )

F. Gourdeau, Z. A. Lykova, M. C. White (2005)

Studia Mathematica

We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(’) ≅ Hₙ()’, where Hₙ()’ is the dual space of the homology group of and Hⁿ(’) is the cohomology group of the dual complex ’. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe explicitly...

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

Zinaida Lykova (2008)

Open Mathematics

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain ⊗ ^ -algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ϕ: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H...

Nouvelles approches de la propriété (T) de Kazhdan

Alain Valette (2002/2003)

Séminaire Bourbaki

Un groupe localement compact G a la propriété (T) de Kazhdan si la 1 -cohomologie de tout G -module hilbertien est nulle. Cette propriété de rigidité de la théorie des représentations de G a trouvé des applications qui vont de la théorie ergodique à la théorie des graphes. Pendant près de 30 ans, les seuls exemples connus de groupes avec la propriété (T), provenaient des groupes algébriques simples sur les corps locaux, ou de leurs réseaux. La situation a radicalement changé ces dernières années :...

On a variant of Kazhdan's property (T) for subgroups of semisimple groups

Mohammed Bachir Bekka, Nicolas Louvet (1997)

Annales de l'institut Fourier

Let Γ be an irreducible lattice in a product G of simple groups. Assume that G has a factor with property (T). We give a description of the topology in a neighbourhood of the trivial one dimensional representation of Γ in terms of the topology of the dual space G ^ of G .We use this result to give a new proof for the triviality of the first cohomology group of Γ with coefficients in a finite dimensional unitary representation.

The proportionality constant for the simplicial volume of locally symmetric spaces

Michelle Bucher-Karlsson (2008)

Colloquium Mathematicae

We follow ideas going back to Gromov's seminal article [Publ. Math. IHES 56 (1982)] to show that the proportionality constant relating the simplicial volume and the volume of a closed, oriented, locally symmetric space M = Γ∖G/K of noncompact type is equal to the Gromov norm of the volume form in the continuous cohomology of G. The proportionality constant thus becomes easier to compute. Furthermore, this method also gives a simple proof of the proportionality principle for arbitrary manifolds.

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