Volume 11, issue 1 (2011)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Algebraic independence of generalized MMM–classes

Johannes Ebert

Algebraic & Geometric Topology 11 (2011) 69–105
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 –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/