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# Homology, Homotopy and Applications

## Volume 22 (2020)

### Number 2

### A simple proof of Curtis’ connectivity theorem for Lie powers

Pages: 251 – 258

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a15

#### Authors

#### Abstract

We give a simple proof of Curtis’ theorem: if $A_{\bullet}$ is a $k$-connected free simplicial abelian group, then $L^n (A_{\bullet})$ is a $k + \lceil \operatorname{log}_2 n \rceil$-connected simplicial abelian group, where $L^n$ is the $n$‑th Lie power functor. In the proof we do not use Curtis’ decomposition of Lie powers. Instead we use the Chevalley–Eilenberg complex for the free Lie algebra.

#### Keywords

homotopy theory, unstable Adams spectral sequence, simplicial group, connectivity, Chevalley–Eilenberg complex

The work is supported by a grant of the Government of the Russian Federation for the state support of scientific research, agreement 14.W03.31.0030 dated 15.02.2018. The third author was also supported by “Native Towns”, a social investment program of PJSC “Gazprom Neft”. 2010 Mathematics Subject Classification: 55Pxx, 55U10, 18G30.

Received 17 December 2019

Accepted 13 January 2020

Published 6 May 2020