Some rational homology computations for diffeomorphisms of odd-dimensional manifolds

TL;DR

This paper computes the rational cohomology of the classifying space of diffeomorphism groups for certain high-dimensional manifolds, revealing a free algebra structure generated by specific characteristic classes.

Contribution

It provides the first explicit calculation of the rational cohomology for these diffeomorphism groups in high dimensions, using a novel combination of classical and modern techniques.

Findings

Rational cohomology is a free graded algebra on Miller–Morita–Mumford classes.

The computation applies to manifolds with large genus g and dimension n.

The method integrates homotopy automorphisms, surgery theory, and algebraic K-theory.

Abstract

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. 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, (c) use pseudoisotopy theory and algebraic $K$-theory to get at actual diffeomorphism groups.