Algebraic & Geometric Topology

Algebraic independence of generalized MMM–classes

Johannes Ebert

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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.

Article information

Algebr. Geom. Topol., Volume 11, Number 1 (2011), 69-105.

Received: 22 March 2010
Revised: 30 June 2010
Accepted: 23 September 2010
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]

characteristic class manifold bundle Miller–Morita–Mumford class


Ebert, Johannes. Algebraic independence of generalized MMM–classes. Algebr. Geom. Topol. 11 (2011), no. 1, 69--105. doi:10.2140/agt.2011.11.69.

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