Difféomorphismes commutants des surfaces et stabilité des fibrations en tores. (Commuting diffeomorphisms of surfaces and stability of fibrations by tori)
The paper contains two partial solutions of Rosenberg's problem concerning the stability of fibres of a fibre bundle by tori over a surface. These are the following: 1) Let S be a compact surface with the non-vanishing Euler characteristic. For each \(n\in {\mathbb{N}}\), there exists a neighbourhood \(\mu_ n\) of the identity in \(Diff_ 1(S)\) such that if \(f_ 1,...,f_ n\in \mu_ n\) are diffeomorphisms of S such that \(f_ if_ j=f_ jf_ i\) for \(i,j=1,...,n\), then they have a common fixed point. 2) Let S be an oriented surface of genus \(g>1\). There exists a fibre bundle over S with fibre \(T_ 2\) and a family \(\{\) \({\mathcal F}_ t\}_{t\in {\mathbb{R}}}\) of foliations depending smoothly on t, such that \({\mathcal F}_ 0\) is defined by a fibre bundle and, for each \(t\neq 0\), \({\mathcal F}_ t\) has no compact leaf.
fibre bundle over an oriented surface with fibre \(T^ 2\)
scientific article; zbMATH DE number 4153490