On fiber-preserving isotopies of surface homeomorphisms. (English) Zbl 0977.57002

Let \(\Sigma_g\) be a closed oriented surface of genus \(g\). Consider a branched covering \(\pi:\Sigma_g\rightarrow S^2\) of the \(2\)-sphere \(S^2\). A homeomorphism \(h\) is called fiber-preserving if \(\pi(x)=\pi(y)\) implies \(\pi(h(x))=\pi(h(y))\). If \(h\) is isotopic to the identity in the class of fiber-preserving maps then we say that \(h\) is fiber-isotopic to the identity. In 1973 J. Birman and H. Hilden proved that if \(\pi\) is a regular covering with a finite group of translations which fix each branch point then every fiber-preserving homeomorphism \(h\), isotopic to the identity, is fiber-isotopic to the identity. In this paper with the help of a special \(3\)-fold covering it is proved that the regularity of \(\pi\) is essential. The proof is based on the technique of symplectic Lefschetz fibrations on \(4\)-manifolds.


57M12 Low-dimensional topology of special (e.g., branched) coverings
57N37 Isotopy and pseudo-isotopy
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