Some rational homology computations for diffeomorphisms of odd‐dimensional manifolds

Abstract

AbstractWe calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds , for large and , up to degree . 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 ‐theory to get at actual diffeomorphism groups.