In their work on Hopf-algebras:
on the last page p.263, they say that the Hurewicz map $\lambda: \pi_*(G)\otimes K \rightarrow H_*(G,K)$ is morphism of Lie-algebras for fields $K$ with characteristic $0$, which is fine. But they claim that the image of this map is excatly $P(H_*(G,K))$ the primitive elements of the Lie-algebra $H_*(G,K)$. For the proof he refers to Cartan and Serre without giving an reference. Does anyone know if the proof is written down anywhere?
Thanks in advance

  • 2
    $\begingroup$ Such an element must come from $H_*(S,K)$, but all elements of $H_*(S,K)$ are primitive. $\endgroup$
    – user43326
    Jun 11, 2019 at 17:59
  • 1
    $\begingroup$ @user43326 that only shows that the Hurewicz image lands in the primitives, not that it hits all of them. That requires an argument! Milnor and Moore sketch an argument by inducting up the Postnikov tower which is nice $\endgroup$ Jun 12, 2019 at 1:02
  • $\begingroup$ Oh, I misread OP. $\endgroup$
    – user43326
    Jun 12, 2019 at 6:14

1 Answer 1


This is immediate from Proposition 9.2.4 in the book ``More concise algebraic topology'' by Kate Ponto and myself. No originality is claimed.


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