A Reducibility Problem for Monodromy of Some Surface Bundles

Abstract

A surface bundle determines a monodromy representation recording the twisting of the fiber under transport around a closed path in the base space. The fascinating relation between these monodromy representations and the Thurston classification of surface automorphisms will be studied. In this note we deal with a simple and interesting case: the fibrations of Fadell and Neuwirth.unless otherwise stated. Let {x 0 1 , x 0 2 , . . .} be a sequence of distinct points of X. Define the surface X n of finite type (g, m + n) by setting1.2) 597 J. Knot Theory Ramifications 2004.13:597-616. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/05/15. For personal use only. a Clearly, Isot(X, n) consists of those elements f ∈ Mod(Xn) that are induced by self maps of Xn that fix each specified puncture x 0 i on Xn and are isotopic to the identity as self maps of X. Isot stands for isotropy subgroup in Mod(Xn). J. Knot Theory Ramifications 2004.13:597-616. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/05/15. For personal use only.