## Geometry & Topology

### Relations among characteristic classes of manifold bundles

Ilya Grigoriev

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.gt/1508437636

Digital Object Identifier
doi:10.2140/gt.2017.21.2015

Mathematical Reviews number (MathSciNet)
MR3654103

Zentralblatt MATH identifier
1370.55005

#### Citation

Grigoriev, Ilya. Relations among characteristic classes of manifold bundles. Geom. Topol. 21 (2017), no. 4, 2015--2048. doi:10.2140/gt.2017.21.2015. https://projecteuclid.org/euclid.gt/1508437636

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