Fuller, Terry On fiber-preserving isotopies of surface homeomorphisms. (English) Zbl 0977.57002 Proc. Am. Math. Soc. 129, No. 4, 1247-1254 (2001). 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. Reviewer: S.R.Nasyrov (Kazan’) Cited in 5 Documents MSC: 57M12 Low-dimensional topology of special (e.g., branched) coverings 57N37 Isotopy and pseudo-isotopy Keywords:Riemann surface; branched covering; isotopy; Lefschetz fibration; Dehn twist × Cite Format Result Cite Review PDF Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.