Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On braid monodromies of non-simple braided surfaces

Seiichi Kamadaa1

a1 Department of Mathematics, Osaka City University, Osaka, 558, Japan

A braided surface of degree m is a compact oriented surface S embedded in a bidisk S030500410007482X_inline1 such that S030500410007482X_inline2 is a branched covering map of degree m and S030500410007482X_inline3, where S030500410007482X_inline4 is the projection. It was defined L. Rudolph [14, 16] with some applications to knot theory, cf. [13, 14, 15, 16, 17, 18]. A similar notion was defined O. Ya. Viro: A (closed) 2-dimensional braid in R4 is a closed oriented surface F embedded in R4 such that S030500410007482X_inline5 and pr2F: FS2 is a branched covering map, where S030500410007482X_inline6 is the tubular neighbourhood of a standard 2-sphere in R4. It is related to 2-knot theory, cf. [8, 9, 10]. Braided surfaces and 2-dimensional braids are called simple if their associated branched covering maps are simple. Simple braided surfaces and simple 2-dimensional braids are investigated in some articles, [5, 8, 9, 14, 16], etc. This paper treats of non-simple braided surfaces in the piecewise linear category. For braided surfaces a natural weak equivalence relation, called braid ambient isotopy, appears essentially although it is not important for classical dimensionai braids Artin's argument [1].

(Received March 09 1995)

(Revised June 13 1995)