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Abstract
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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.
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Keywords
characteristic class, manifold bundle,
Miller–Morita–Mumford class
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Mathematical Subject Classification 2000
Primary: 55R40
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Publication
Received: 22 March 2010
Revised: 30 June 2010
Accepted: 23 September 2010
Published: 6 January 2011
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