Some Canonical Cohomology Classes on Groups of Volume Preserving Diffeomorphisms

Abstract

We discuss some canonical cohomology classes on the space $\bar B\mathcal {D}iff_{\omega 0}^cM$, where $\mathcal {D}iff_{\omega 0}^cM$ is the identity component of the group of compactly supported diffeomorphisms of the manifold $M$ which preserve the volume form $\omega$. We first look at some classes ${c_k}(M),1 \leqslant k \leqslant n = {\text {dim}} M$, which are defined for all $M$, and show that the top class ${c_n}(M) \in {H^n}(\bar B\mathcal {D}iff_{\omega 0}^cM;{\mathbf {R}})$ is nonzero for $M = {S^n},n$ odd, and is zero for $M = {S^n},n$ even. When $H_c^i(M;{\mathbf {R}}{\text {) = 0}}$ for $0 \leqslant i < n$, the classes ${c_k}(M)$ all vanish and a secondary class $s(M) \in {H^{n - 1}}(\bar B\mathcal {D}iff_{\omega 0}^cM; {\mathbf {R}})$ may be defined. This is trivially zero when $n$ is odd, and is twice the Calabi invariant for symplectic manifolds when $n = 2$. We prove that $s({{\mathbf {R}}^n}) \ne 0$ when $n$ is even by showing that it is one of a set of nonzero classes which were defined by Hurder in [7].