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.

中文翻译：

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关于流形自构的理性同构稳定性的一个注记

通过Berglund和Madsen的工作，纤维| $ D ^ {2n} \ sharp（S ^ n \ times S ^ n）^ {\ sharp g} $ |⁠的纤维和光滑块束的理性特征类别的环，相对于边界，为| $ 2n \ ge 6 $ | 独立于| $ g $ | 以度| $ * \ le（g-6）/ 2 $ |⁠为单位。在本说明中，我们解释了如何将该范围提高到| $ * \ le g-2 $ | 由于Borel和经典不变理论而使用同调消失结果。这意味着光滑束的类似环独立于| $ g $ | 在相同范围内，条件是度数与尺寸相比较小。