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 $D^{2n}♯(S^n\times S^n{)}^{♯g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $\ast \le (g-6)/2$. In this note, we explain how this range can be improved to $\ast \le g-2$ 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.

arXiv comment8 pages, to appear in The Quarterly Journal of Mathematics

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Cited by in corpus (1)

• The stable cohomology of self-equivalences of connected sums of products of spheres