On the divisibility of characteristic classes of non-oriented surface bundles

TL;DR

This paper introduces a construction for fiberwise orientation coverings of manifold bundles and demonstrates that zeta classes of unoriented surface bundles are not divisible in the stable range.

Contribution

It presents a novel construction linking orientation coverings to characteristic classes, revealing non-divisibility of zeta classes in unoriented surface bundles.

Findings

Zeta classes are not divisible in the stable range.

A new construction for fiberwise orientation coverings.

Implications for characteristic classes of surface bundles.

Abstract

In this note we introduce a construction which assigns to an arbitrary manifold bundle its fiberwise orientation covering. This is used to show that the zeta classes of unoriented surface bundles are not divisible in the stable range.