Stable characteristic classes of smooth manifold bundles

TL;DR

This paper investigates stable characteristic classes of smooth oriented manifold bundles, showing that in even dimensions, all such classes are tautological and generated by Miller-Morita-Mumford classes.

Contribution

It establishes that rational stable characteristic classes in even dimensions are tautological, extending the understanding of characteristic classes from vector bundles to manifold bundles.

Findings

All rational stable characteristic classes in even dimensions are tautological.

Stable classes are generated by Miller-Morita-Mumford classes in even dimensions.

Extension of classes to homotopy colimits characterizes stability.

Abstract

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Similarly, characteristic classes of smooth oriented manifold bundles with fibers given by oriented closed smooth manifolds of a fixed dimension d\ge 0 can be identified with cohomology classes of the disjoint union of classifying spaces BDiff M of orientation preserving diffeomorphism groups of oriented closed manifolds of dimension d. A characteristic class is stable if it extends to a cohomology class of a homotopy colimit of spaces BDiff M. We show that each rational stable characteristic class of oriented manifold bundles of even dimension d is tautological, e.g., if d=2, then each rational stable characteristic class is a polynomial in terms of Miller-Morita-Mumford classes.