A Simple proof of Curtis' connectivity theorem for Lie powers

Abstract

We give a simple proof of the Curtis' theorem: if $A_{\bullet }$ is $k$-connected free simplicial abelian group, then $L^n(A_{\bullet })$ is an $k+\lceil {\log }_{2}n\rceil$-connected simplicial abelian group, where $L^n$ is the functor of $n$-th Lie power. In the proof we do not use Curtis' decomposition of Lie powers. Instead of this we use the Chevalley-Eilenberg complex for the free Lie algebra.