an:02209042
Zbl 1080.57021
Imayoshi, Yoichi; Ito, Manabu; Yamamoto, Hiroshi
A reducibility problem for monodromy of some surface bundles
EN
J. Knot Theory Ramifications 13, No. 5, 597-616 (2004).
00108829
2004
j
57M99 57M50 55S37 20F36 57N05
surface bundles; monodromy; Thurston's classification of surface automorphisms; Teichm??ller space; braid groups; Fadell-Neuwirth sequence
Let \(X\) denote an orientable closed surface of genus \(g\) with \(n\) points removed. \textit{E. Fadell} and \textit{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 Teichm??ller 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.
Daciberg Gon??alves (S??o Paulo)
Zbl 0136.44104