A note on rational homological stability for automorphisms of manifolds

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 $<em>\le (g-6)/2$. In this note, we explain how this range can be improved to $</em>\le g-2$ 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.