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A reducibility problem for monodromy of some surface bundles. (English) Zbl 1080.57021
Let \(X\) denote an orientable closed surface of genus \(g\) with \(n\) points removed. E. Fadell and L. Neuwirth in [Math. Scand. 10, 111–118 (1962; Zbl 0136.44104)] defined for each integer \(n\geq 0\) a fibration \(F_1X_n \to F_{n+1}X \to F_nX\) where \(X_n=X\setminus \{x_1^0,\dots x_n^0\}\) and \(F_n\) denotes the \(n\)th configuration space. This fibration naturally defines a monodromy \(\pi_1(F_nX)\to Iso(X,n)\), where \(Iso(X,n)\) is the group of isotopy classes of orientation preserving homeomorphisms \(f:X_n \to X_n\). This work describes a relation between elements \(\beta\in\pi_1(F_nX)\) and their image \(f\in Iso(X_n)\) with respect to the Thurston classification of the surface automorphisms. More precisely using the notation above the authors show:
Theorem: Let \(X\) be an oriented surface of non-excluded finite type \((g,m)\); that is, \(2g-2+m>0\). Then the element \(1\neq f\in Iso(X,n)\) is not of finite order. Further, \(f\) is reducible if and only if it can be induced by \(\beta\in \pi_1(F_nX)\) satisfying at least one of the following conditions: (i) \(\beta\) is non-spreading; (ii) \(\beta\) has a boundary partition; (iii) \(\beta\) has a tube structure over some subset in \(\{1,\dots,n\}\) which is not a singleton.
One crucial step is to show, with the help of Teichmüller theory, that a mapping class \(f\in Iso(X,n)\) is reducible if it can be induced by a braid \(\beta\in \pi_1(F_nX)\) satisfying none of the conditions (i), (ii) and (iii). The paper is well organized and it provides several intuitive geometrical interpretations.

57M99 General low-dimensional topology
57M50 General geometric structures on low-dimensional manifolds
55S37 Classification of mappings in algebraic topology
20F36 Braid groups; Artin groups
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
Full Text: DOI
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