THE CENTER OF A GRADED CONNECTED LIE-ALGEBRA IS A NICE IDEAL

Let (L(V),d) be a free graded connected differential Lie algebra over the field Q of rational numbers. An ideal I in the Lie algebra H(L(V), d) is called nice if, for every cycle alpha is an element of L(V) such that [alpha] belongs to I, the kernel of the map H(L(V),d) --> H(L(V + Qx),d), d(x) = alpha, is contained in I. We show that the center of H(L(V),d) is a nice ideal and we give in that case some informations o n the structure of the Lie algebra H(L(V + Qx),d). We apply this compu tation for the determination of the rational homotopy Lie algebra L(X) = pi(Omega X) X Q of a simply connected space X. We deduce that the kernel of the map L(X) --> L(Y) induced by the attachment of a cell al ong an element in the center is contained in the center.