# A note on rational homological stability for automorphisms of manifolds

## TL;DR

This paper improves the known stability range for rational characteristic classes of certain manifold bundles, showing they are independent of the genus in a larger degree range using cohomological vanishing results.

## Contribution

It extends the stability range for rational characteristic classes of manifold bundles from    to  , using cohomological vanishing and invariant theory.

## Key findings

- Stability range improved to degree  
- Rings are independent of genus in the new range
- Results apply to smooth bundles with small degree relative to dimension

## Abstract

By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp(S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.