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The computation of Stiefel-Whitney classes

Pierre Guillot (2010)

Annales de l’institut Fourier

The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here “compute” means to find a presentation in terms of generators and relations, and involves only the underlying (graded) ring. We propose a method to determine some of the extra structure: namely, Stiefel-Whitney classes and Steenrod operations. The calculations are explicitly carried out for about one hundred groups (the results can be consulted on the Internet).Next,...

The Mumford conjecture

Geoffrey Powell (2004/2005)

Séminaire Bourbaki

The Mumford Conjecture asserts that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra on the Mumford-Morita-Miller characteristic classes; this can be reformulated in terms of the classifying space B Γ derived from the mapping class groups. The conjecture admits a topological generalization, inspired by Tillmann’s theorem that B Γ admits an infinite loop space structure after applying Quillen’s plus construction. The text presents the proof by Madsen and...

The Weil algebra and the Van Est isomorphism

Camilo Arias Abad, Marius Crainic (2011)

Annales de l’institut Fourier

This paper belongs to a series of papers devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman’s BRST model, here we introduce the Weil algebra W ( A ) associated to any Lie algebroid A . We then show that this Weil algebra is related to the Bott-Shulman complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual...

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