Algebraic independence of generalized Morita-Miller-Mumford classes

TL;DR

This paper investigates the algebraic independence of generalized Morita-Miller-Mumford classes in manifold bundles, revealing non-vanishing properties in even dimensions and exceptions in odd dimensions, including holomorphic bundles.

Contribution

It establishes the non-vanishing of all MMM-classes in rational cohomology for even-dimensional manifolds and identifies the unique zero class in odd dimensions, extending to holomorphic bundles.

Findings

All MMM-classes are nonzero in even dimensions for some bundle.

In odd dimensions, all MMM-classes are nonzero except the Hirzebruch $\\\ ext{L}$-class.

The result applies similarly to holomorphic fibre bundles.

Abstract

The generalized Morita-Miller-Mumford classes of a smooth oriented manifold bundle are defined as the image of the characteristic classes of the vertical tangent bundle under the Gysin homomorphism. We show that if the dimension of the manifold is even, then all MMM-classes in rational cohomology are nonzero for some bundle. In odd dimensions, this is also true with one exception: the MMM-class associated with the Hirzebruch $\cL$-class is always zero. We also show a similar result for holomorphic fibre bundles.