On the continuous cohomology of diffeomorphism groups

TL;DR

This paper establishes monomorphisms from certain cohomology groups related to the Weil algebra into the continuous cohomology of diffeomorphism groups of manifolds, revealing new connections between geometric structures and group cohomology.

Contribution

It proves the existence of monomorphisms from $H^m(W_n,O(n))$ into the continuous cohomology of diffeomorphism groups under specific conditions, extending understanding of their cohomological properties.

Findings

Monomorphism from $H^m(W_n,O(n))$ to $H^m_{cont}(Diff M, R)$ for manifolds with trivial higher cohomology.

Monomorphism from $H^m(W_n,O(n))$ to $H^{m-n}_{cont}(Diff_+ M, R)$ for closed oriented manifolds.

Results connect algebraic cohomology of Lie algebras with the topology of diffeomorphism groups.

Abstract

Suppose that $M$ is a connected orientable $n$-dimensional manifold and $m>2n$. If $H^i(M,\R)=0$ for $i>0$, it is proved that for each $m$ there is a monomorphism $H^m(W_n,\on{O}(n))\to H^m_{\on{cont}}(\on{Diff}M,\R)$. If $M$ is closed and oriented, it is proved that for each $m$ there is a monomorphism $H^m(W_n,\on{O}(n))\to H^{m-n}_{\on{cont}}(\on{Diff}_+M,\R)$, where $\on{Diff}_+M$ is a group of preserving orientation diffeomorphisms of $M$.