  Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm.
Sup. (4) 24 (6)(1991) 635) found a Lie bialgebra structure on the vector space
generated by non-trivial free homotopy classes of loops on an orientable
surface. Chas (Combinatorial Lie bialgebras of curves on surfaces, Topology 43
(2004) 543), by the aid of the computer, found a negative answer to Turaev's
question about the characterization of multiples of simple classes in terms of
the cobracket, in every surface of negative Euler characteristic and positive
genus. However, she left open Turaev's conjecture, namely if, for genus zero,
every class with cobracket zero is a multiple of a simple class. The aim of
this paper is to give a positive answer to this conjecture.
