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arXiv · 1310.6804

Relations among characteristic classes of manifold bundles

Ilya Grigoriev
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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 $S^d \times S^d$ and an odd $d$, we find numerous algebraic relations among the 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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BibTeXRIS

Ilya Grigoriev. 2016-07-18. Relations among characteristic classes of manifold bundles. https://doi.org/10.2140/gt.2017.21.2015

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