Configuration spaces, moduli spaces and 3-fold covering spaces

초록

We have, in this paper, constructed a newnon-geometric embedding of some braid group into the mapping class group of a surface which is induced by the 3-fold branched covering over a disk with some branch points. There is a lift (beta) over tilde (i) of the half-Dehn twist (beta) over tilde (i) on the disk with some marked points to some surface via the 3-fold covering. We show how this lift (beta) over tilde (i) acts on the fundamental group of the surface, and also show that (beta) over tilde (i) equals the product of two (inverse) Dehn twists. Two adjacent lifts satisfy the braid relation, hence such lifts induce a homomorphism phi : B-k -> Gamma(g,b). In this paper we give a concrete description of this homomorphism and show that it is injective by the Birman-Hilden theory. Furthermore, we show that the map on the level of classifying spaces of groups is compatible with the action of little 2-cube operad so that it induces a trivial homomorphism on the stable homology.

키워드