We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:={\#}^g(S^n\times S^{n+1})\setminus \mathrm{int}(D^{2n+1})$, for large $g$ and $n$, up to degree $n-3$. The answer is that it is a free graded commutative algebra on an appropriate set of Miller–Morita–Mumford classes. Our proof goes through the classical three-step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, and (c) use pseudoisotopy theory and algebraic $K$-theory to get at actual diffeomorphism groups.

中文翻译：

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奇维流形微分同胚的一些有理同调计算

我们计算流形微分同胚群分类空间的有理上同调$U_{G,1}^n:={\#}^G（S^n\times S^{n+1}）\setminus \mathrm{整数}（D^{2n+1}）$，对于大$G$和$n$, 达到程度$n-3$。答案是，它是一组适当的 Miller-Morita-Mumford 类上的自由分级交换代数。我们的证明经历了经典的三步过程：（a）计算同伦自同构的上同调，（b）使用手术将其与块微分同胚进行比较，以及（c）使用伪同位素理论和代数$K$-获得实际微分同胚群的理论。