A vanishing theorem for tautological classes of aspherical manifolds

Hebestreit, Fabian¹ ; Land, Markus² ; Lück, Wolfgang¹ ; Randal-Williams, Oscar³

¹ Mathematical Institute, University of Bonn, Bonn, Germany
² Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark
³ Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom

Geometry & topology, Tome 25 (2021) no. 1, pp. 47-110.

Résumé

Tautological classes, or generalised Miller–Morita–Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.

Détail

DOI : 10.2140/gt.2021.25.47

Classification : 55R20, 55R40, 55R60, 57P10
Keywords: aspherical closed manifolds, tautological classes, characteristic classes, manifold bundles, Burghelea's conjecture

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