The braidings of mapping class groups and loop spaces

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