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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Relations among characteristic classes of manifold bundles

Ilya Grigoriev

Geometry & Topology 21 (2017) 2015–2048
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 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.

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/