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A vanishing theorem for tautological classes of aspherical manifolds

Fabian Hebestreit, Markus Land, Wolfgang Lück and Oscar Randal-Williams

Geometry & Topology 25 (2021) 47–110
Abstract

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.

Keywords
aspherical closed manifolds, tautological classes, characteristic classes, manifold bundles, Burghelea's conjecture
Mathematical Subject Classification 2010
Primary: 55R20, 55R40, 55R60, 57P10
References
Publication
Received: 30 July 2018
Revised: 26 November 2019
Accepted: 17 January 2020
Published: 2 March 2021
Proposed: Ulrike Tillmann
Seconded: Haynes R Miller, Ralph Cohen
Authors
Fabian Hebestreit
Mathematical Institute
University of Bonn
Bonn
Germany
Markus Land
Department of Mathematical Sciences
University of Copenhagen
Copenhagen
Denmark
Wolfgang Lück
Mathematical Institute
University of Bonn
Bonn
Germany
Oscar Randal-Williams
Centre for Mathematical Sciences
University of Cambridge
Cambridge
United Kingdom