Detecting and realising characteristic classes of manifold bundles

TL;DR

This paper explores the non-triviality and dependence of characteristic classes in manifold bundles, extending the higher-dimensional Mumford conjecture and analyzing how these classes relate to characteristic numbers of fibers, total spaces, and bases.

Contribution

It applies advanced theoretical work to prove non-triviality of characteristic classes and investigates their dependence on characteristic numbers, advancing understanding in manifold bundle topology.

Findings

Proved non-triviality of certain characteristic classes

Analyzed dependence of classes on characteristic numbers

Extended higher-dimensional Mumford conjecture insights

Abstract

We apply our earlier work on the higher-dimensional analogue of the Mumford conjecture to two questions. Inspired by work of Ebert we prove non-triviality of certain characteristic classes of bundles of smooth closed manifolds. Inspired by work of Church-Farb-Thibault and Church-Crossley-Giansiracusa we investigate the dependence of characteristic classes of bundles on characteristic numbers of its fibre, total space and base space.