Geometry & Topology

Relations among characteristic classes of manifold bundles

Ilya Grigoriev

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We study relations among characteristic classes of smooth manifold bundles with highly connected fibers. For bundles with fiber the connected sum of g copies of a product of spheres Sd × Sd, where d is odd, we find numerous algebraic relations among so-called “generalized Miller–Morita–Mumford classes”. For all g > 1, we show that these infinitely many classes are algebraically generated by a finite subset.

Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with g, according to recent homological stability results. In the case of surface bundles (d = 1), our approach recovers some previously known results about the structure of the classical “tautological ring”, as introduced by Mumford, using only the tools of algebraic topology.

Article information

Geom. Topol., Volume 21, Number 4 (2017), 2015-2048.

Received: 30 October 2013
Revised: 25 May 2016
Accepted: 8 July 2016
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 55T10: Serre spectral sequences 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx]

manifold bundles characteristic classes tautological ring Miller–Morita–Mumford classes


Grigoriev, Ilya. Relations among characteristic classes of manifold bundles. Geom. Topol. 21 (2017), no. 4, 2015--2048. doi:10.2140/gt.2017.21.2015.

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