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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some canonical cohomology classes on groups of volume preserving diffeomorphisms


Author: Dusa McDuff
Journal: Trans. Amer. Math. Soc. 275 (1983), 345-356
MSC: Primary 58H10; Secondary 57R50, 57T99, 58D05
MathSciNet review: 678355
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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].


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0678355-7
PII: S 0002-9947(1983)0678355-7
Article copyright: © Copyright 1983 American Mathematical Society