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Three cocycles on \(\text{Diff}(S^1)\) generalizing the Schwarzian derivative. (English) Zbl 0919.57026

Let \(\text{Diff} (\mathbb{R} P^1)\) be the group of diffeomorphisms of the circle \(\mathbb{R} P^1=S^1\). Denote by \(\mathcal{D}_{\lambda ,\mu }^{k}\), \(\lambda ,\mu \in \mathbb{R}\) the space of differential operators \(\mathcal{A}=a_{k}(x)\frac{d^{k}} {dx^{k}} +\ldots +a_{0}(x)\) with \(\text{Diff} (\mathbb{R} P^{1})\)-action: \( f_{\lambda ,\mu }(\mathcal{A})=f_{\lambda }^{\ast }\circ \mathcal{A}\circ f_{\mu }^{\ast }\), where \(f\in \text{Diff}(\mathbb{R} P^{1})\) and \(f_{\lambda }^{\ast }(\phi)=\phi \circ f^{-1}\cdot (f^{-1'})^{\lambda }\). In this paper the action of the group \(\text{Diff} (\mathbb{R} P^1)\) on this space \(\mathcal{D}_{\lambda ,\mu }^{k}\) is considered and the differentiable cohomology of the group \(\text{Diff} (\mathbb{R} P^1) \) with coefficients in the module \(\mathcal{D}_{\lambda,\mu}\), vanishing on \(\text{PSL} (2,\mathbb{R})\), is studied. The main results of this paper are
(1) For the generic values of \(\lambda\), one has \(H^1(\text{Diff}(\mathbb{R} P^1),\text{PSL} (2,\mathbb{R});\mathcal{D}_{\lambda,\mu})=\mathbb{R}\), if \(\mu -\lambda=2,3,4\) and \(0\), otherwise.
(2) The nontrivial \(\text{PSL} (2,\mathbb{R})\)-invariant cocycles \(\mathcal{S}_{\lambda}\), \(\mathcal{T}_{\lambda}\), \(\mathcal{U}_{\lambda}\) with values in the modules \(\mathcal{D} _{\lambda,\lambda +2}\), \(\mathcal{D}_{\lambda,\lambda +3}\), \(\mathcal{D} _{\lambda,\lambda +4}\) respectively, are given by the formulae: \(\mathcal{S}_{\lambda}(f)=S(f)\), \(\mathcal{T}_{\lambda}(f)= S(f)\frac {d} {dx}-\frac{\lambda} {2}S(f)'\), \(\mathcal{U}_{\lambda}= S(f)\frac {d^2} {dx^2}-\frac{2\lambda+1} {2}S(f)'\frac {d} {dx}+ \frac{ \lambda(2\lambda+1)} {10}S(f)''- \frac{\lambda(\lambda+3)} {5} S(f)^2\), where \(S(f)\) is the Schwarzian derivative. These cocycles are nontrivial for every \(\lambda \neq -1/2, -1,-3/2\), respectively.

MSC:

57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
57T10 Homology and cohomology of Lie groups
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
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