Some rational homology computations for diffeomorphisms of odd‐dimensional manifolds.

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds Ug,1n:=#g(Sn×Sn+1)∖int(D2n+1)$U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}(D^{2n+1})$, for large g$g$ and n$n$, up to degree n−3$n-3$. 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$K$‐theory to get at actual diffeomorphism groups. [ABSTRACT FROM AUTHOR]