A Simple proof of Curtis' connectivity theorem for Lie powers

TL;DR

This paper presents a simplified proof of Curtis' connectivity theorem for Lie powers of free simplicial abelian groups, avoiding complex decompositions by utilizing the Chevalley-Eilenberg complex.

Contribution

It provides a more straightforward proof of Curtis' theorem, removing the need for Curtis' decomposition and employing the Chevalley-Eilenberg complex.

Findings

Proves that $L^n(A_\bullet)$ increases connectivity by $\lceil \log_2 n \rceil$

Simplifies the proof of Curtis' theorem

Avoids Curtis' decomposition in the proof

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.