Algebraic independence of generalized MMM–classes

Ebert, Johannes

doi:10.2140/agt.2011.11.69

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Algebraic independence of generalized MMM–classes

Johannes Ebert

Algebraic & Geometric Topology 11 (2011) 69–105

DOI: 10.2140/agt.2011.11.69

Abstract

The generalized Miller–Morita–Mumford classes (MMM classes) of a smooth oriented manifold bundle are defined as the image of the characteristic classes of the vertical tangent bundle under the Gysin homomorphism. We show that if the dimension of the manifold is even, then all MMM–classes in rational cohomology are nonzero for some bundle. In odd dimensions, this is also true with one exception: the MMM–class associated with the Hirzebruch $ℒ$ –class is always zero. Moreover, we show that polynomials in the MMM–classes are also nonzero. We also show a similar result for holomorphic fibre bundles and for unoriented bundles.

Keywords

characteristic class, manifold bundle, Miller–Morita–Mumford class

Mathematical Subject Classification 2000

Primary: 55R40

References

Publication

Received: 22 March 2010

Revised: 30 June 2010

Accepted: 23 September 2010

Published: 6 January 2011

Authors

Johannes Ebert
Mathematisches Institut Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany
http://www.math.uni-bonn.de/people/ebert/

Volume 11, issue 1 (2011)