Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

Pub Date : 2024-01-31 DOI:10.1112/topo.12324

Johannes Ebert, Jens Reinhold

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Abstract

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g, 1}^{n} : = #^{g} (S^{n} \times S^{n + 1}) ∖ int (D^{2 n + 1})$ , 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.

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奇维流形差分同调的一些理性同调计算

我们计算流形差分群分类空间的有理同调 U g , 1 n : = # g ( S n × S n + 1 ) ∖ int ( D 2 n + 1 ) $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}(D^{2n+1})$，对于大g $g$和n $n$，直到度数n - 3 $n-3$。答案是，它是一个在适当的米勒-莫里塔-蒙福德类集合上的自由分级交换代数。我们的证明经历了经典的三步程序：(a) 计算同调自形体的同调；(b) 利用外科手术将其与块差形体进行比较；(c) 利用伪拟态理论和代数 K $K$ 理论得到实际的差形群。

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