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

Sergei O. Ivanov (Laboratory of Modern Algebra and Applications, St. Petersburg State University, Saint Petersburg, Russia)

Vladislav Romanovskii (Laboratory of Modern Algebra and Applications, St. Petersburg State University, Saint Petersburg, Russia)

Andrei Semenov (Chebyshev Laboratory, St. Petersburg State University, Saint Petersburg, Russia)

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

Received 17 December 2019

Accepted 13 January 2020

Published 6 May 2020