Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

Abstract

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