The action of image of braiding under the Harer map

Abstract

John Harer conjectured that the canonical map from braid group tomapping class group induces zero homology homomorphism. To provethe conjecture it suffices to show that this map preserves thefirst Araki-Kudo-Dyer-Lashof operation. To get information on thishomology operation we need to investigate the image of braidingunder the Harer map. The main result of this paper is to give bothalgebraic and geometric interpretations of the image of braidingunder the Harer map. For this we need to calculate long chains ofconsecutive actions of Dehn twists on the fundamental group ofsurface.