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Let $R \to R'$ be a morphism of connected free simplicial Lie algebras. Then the analog of the Whitehead lemma states that if $\pi_* f_{\mathrm{ab}}$ is an isomorphism then $\pi_* f$ is an isomorphism. I am looking for the proof of this statement. References that are known to me are the six authors paper on the unstable Adams spectral sequence and "Simplicial homotopy theory" by E. Curtis, but in these references the statement relies on the lemma that I do not know how to prove:

Lemma If $R$ is a connected free simplicial Lie algebra, then $\pi_q \gamma_r R = 0$ for $q < \log_2 r$.

Here $\gamma_rR$ is the $r$-th term of the lower central series of $R$.

It is claimed that it is sufficient to consider the case when $R = L(M)$, i.e. when $R$ is a free Lie algebra generated by some simplicial module $M$. In this case the statement is indeed clear, but I do not understand why it is sufficient to consider this case. The similar argument for groups works since we have a classifying space functor $\overline W$ and hence every free simplicial group $F$ is homotopy equivalent to $G\overline W F$, where $G$ is a Kan loop group functor. But we do not have a classifying space functor for simplicial Lie algebras, so I am a bit confused.

Any references and ideas are appreciated.

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  • $\begingroup$ arxiv.org/abs/1912.03086 Here is a very clear and conceptual proof of Curtis lemma. $\endgroup$
    – Denis T
    Commented Apr 17 at 10:20
  • $\begingroup$ @DenisT thank you for the link. This paper proves the connectivity for Lie powers, i.e. the functors $\gamma_n / \gamma_{n+1}$. I would like to understand how to derive the connectivity of $\gamma_n$ from this. I thought it would be enough to consider a homotopy exact sequence for $\gamma_{n+1} \to \gamma_n \to \gamma_n / \gamma_{n+1}$, but apparently it is not. I asked one of the authors of the abovementioned paper personally, he also do not know the answer. $\endgroup$
    – thrw
    Commented Apr 18 at 8:27

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