A NOTE ON RATIONAL HOMOLOGICAL STABILITY OF AUTOMORPHISMS OF MANIFOLDS

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 the 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.