Crossed simplicial groups of framed braids and mapping class groups of surfaces (Q1366364)

It is well known that the geometric realization of a cyclic crossed simplicial group is homeomorphic to the circle. The author associates with a surface the crossed simplicial groups \(FB^P_\bullet(M)\) and \(FMC^P_\bullet(M)\) of framed braids and framed mapping class groups. Then their homotopy type is calculated and, in particular, isomorphisms \(FB^P_\bullet(S^2)\simeq SU(2)\) and \(FMC^P_\bullet(S^2)\simeq SO(3)\) are deduced [cf. the author, 34-th Conf. of Lith. Math. Soc., Vilnius, 53-54 (1993)].