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Relations among characteristic classes of manifold bundles

Ilya Grigoriev

Geometry & Topology 21 (2017) 2015–2048

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.

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manifold bundles, characteristic classes, tautological ring, Miller–Morita–Mumford classes
Mathematical Subject Classification 2010
Primary: 55R40, 55T10, 57R22
Received: 30 October 2013
Revised: 25 May 2016
Accepted: 8 July 2016
Published: 19 May 2017
Proposed: Shigeyuki Morita
Seconded: Ralph Cohen, Stefan Schwede
Ilya Grigoriev
Department of Mathematics
University of Chicago
5734 S University Ave
Chicago, IL 60637
United States