Classe d'Euler et minimal exceptionnel
Let \(\Sigma\) be a compact oriented surface different from \(S^ 2\). For a representation \(\phi\) : \(\pi\) \({}_ 1(\Sigma)\to Diff^+(S^ 1)\) let eu(\(\phi)\) denote the Euler class of the fibre bundle obtained by suspension of \(\phi\) evaluated on the fundamental class of \(\Sigma\). It is known that \(| eu(\phi)| \leq | \chi (\Sigma)|.\) The author shows that the inequality is sharp if \(\phi\) has a minimal exceptional set and takes on its values in the subgroup \(Diff^+_ 2(S^ 1)\) of \(C^ 2\)-diffeomorphisms. If, moreover, \(\phi\) is \(Diff^+_{\omega}(S^ 1)\)-valued (class \(C^{\omega})\), then \(eu(\phi)=0\). If \(eu(\phi)=\pm \chi (\Sigma)\) for a \(\phi\) : \(\pi\) \({}_ 1(\Sigma)\to Homeo^+(S^ 1)\) then every leaf of the suspended codimension 1 foliation is either a cylinder, or a plane, or a torus.
scientific article; zbMATH DE number 3982089