The braidings of mapping class groups and loop spaces

Abstract

The disjoint union of mapping class groups forms a braided monoidal category. We give an explicit expression of braidings in terms of both their actions on the fundamental group of the surface and the standard Dehn twists. This braided monoidal category gives rise to a double loop space. We prove that the action of little 2-cube operad does not extend to the action of little 3-cube operad by showing that the Browder operation induced by 2-cube operad action is nontrivial. A rather simple expression of Reshetikhin-Turaev representation is given for the sixteenth root of unity in terms of matrices with entries of complex numbers. We show by matrix calculation that this representation is symmetric with respect to the braid structure.