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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.

##### MSC:
 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)
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##### References:
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