Cohomology of the diffeomorphism group of the connected sum of two generic lens spaces

TL;DR

This paper computes the rational cohomology ring of the classifying space of the diffeomorphism group of the connected sum of two generic lens spaces, revealing new insights into their topological structure.

Contribution

It provides the first explicit calculation of the cohomology ring for the diffeomorphism group of connected sums of generic lens spaces.

Findings

Cohomology ring $H^*(B\text{Diff}(L_1\#L_2);\mathbb{Q})$ computed

Utilizes known homotopy types of diffeomorphism groups of lens spaces

Employs Hatcher's theorem to establish the result

Abstract

We consider the connected sum of two three-dimensional lens spaces $L_1\#L_2$, where $L_1$ and $L_2$ are non-diffeomorphic and are of a certain "generic" type. Our main result is the calculation of the cohomology ring $H^\ast(B\text{Diff}(L_1\#L_2);\mathbb{Q})$, where $\text{Diff}(L_1\#L_2)$ is the diffeomorphism group of $M$ equipped with the $C^\infty$-topology. We know the homotopy type of the diffeomorphism groups of generic lens spaces this, combined with a theorem of Hatcher forms the basis of our argument.