Relations among characteristic classes of manifold bundles

Grigoriev, Ilya

doi:10.2140/gt.2017.21.2015

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Relations among characteristic classes of manifold bundles

Ilya Grigoriev

Geometry & Topology 21 (2017) 2015–2048

DOI: 10.2140/gt.2017.21.2015

Abstract

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 $S^{d} \times S^{d}$ , 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 55R40, 55T10, 57R22

References

Publication

Received: 30 October 2013

Revised: 25 May 2016

Accepted: 8 July 2016

Published: 19 May 2017

Proposed: Shigeyuki Morita

Seconded: Ralph Cohen, Stefan Schwede

Authors

Ilya Grigoriev
Department of Mathematics University of Chicago 5734 S University Ave Chicago, IL 60637 United States
http://math.uchicago.edu/~ilyagr/

Volume 21, issue 4 (2017)