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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.

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
57N37 Isotopy and pseudo-isotopy
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[1] Israel Berstein and Allan L. Edmonds, On the construction of branched coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87 – 124. · Zbl 0359.55001
[2] Joan S. Birman and Hugh M. Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. (2) 97 (1973), 424 – 439. · Zbl 0237.57001 · doi:10.2307/1970830
[3] T. Fuller, Hyperelliptic Lefschetz fibrations and branched covering spaces, to appear, Pacific J. Math. · Zbl 0971.57027
[4] R. Gompf and A. Stipsicz, An introduction to 4-manifolds and Kirby calculus, book in preparation. · Zbl 0933.57020
[5] Hugh M. Hilden, Three-fold branched coverings of \?³, Amer. J. Math. 98 (1976), no. 4, 989 – 997. · Zbl 0342.57002 · doi:10.2307/2374037
[6] H. Hilden, personal communication.
[7] A. Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math. 89 (1980), no. 1, 89 – 104. · Zbl 0457.14011
[8] José M. Montesinos, Three-manifolds as 3-fold branched covers of \?³, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85 – 94. · Zbl 0326.57002 · doi:10.1093/qmath/27.1.85
[9] José María Montesinos, 4-manifolds, 3-fold covering spaces and ribbons, Trans. Amer. Math. Soc. 245 (1978), 453 – 467. · Zbl 0359.55002
[10] B. Siebert and G. Tian, On hyperelliptic \(C^{\infty}\)-Lefschetz fibrations of four-manifolds, Commun. Contemp. Math. 1 (1999), 255-280. CMP 99:14 · Zbl 0948.57018
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