  We establish certain smash operations on spaces over operads which are
general analogues of the Samelson product on single loop spaces, and obtain a
conceptual description of the structure of the homotopy groups of spaces $Y$
over a symmetric $K(\pi,1)$ operad: $\pi_*Y$ is a module over the free
algebraic symmetric operad generated by operations on homotopy groups induced
by these smash operations. In particular the homotopy groups of double loop
spaces is a module over the free algebraic symmetric operad generated by the
conjugacy classes of Brunnian braids modulo the conjugation action of pure
braids.
