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A theorem of W. Browder [1] from 1961 says that an H-space $X$ with finitely generated integral homology (meaning that $H_*(X)$ is finitely generated) has $\pi_2(X) = 0$. This generalizes Cartan's famous theorem that $\pi_2$ of a Lie group vanishes.

I would like to know if there is a relative version of the following sort: Suppose that $f: X \to Y$ is an H-space map and $\ker(H_*(X) \to H_*(Y))$ is finitely generated. Then can I conclude that $\ker(\pi_2(X) \to \pi_2(Y))$ is $0$?

[1] Torsion in H-Spaces, Annals of Mathematics, Second Series, Vol. 74, No. 1 (Jul., 1961), pp. 24-51. http://www.jstor.org/stable/1970305 .

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  • $\begingroup$ $X=K(\mathbb{Z},2)$ is a counterexample to your first paragraph. Did you forget an assumption? $\endgroup$
    – archipelago
    Commented Nov 30, 2022 at 22:44
  • $\begingroup$ @archipelago: "finitely generated" refers to the totality of the integral homology groups. $\endgroup$
    – Oscar Randal-Williams
    Commented Nov 30, 2022 at 23:10
  • $\begingroup$ Makes sense. Thanks. $\endgroup$
    – archipelago
    Commented Nov 30, 2022 at 23:27
  • $\begingroup$ @archipelago Sorry for the confusion; I've edited to clarify "finitely generated". $\endgroup$
    – Danny Ruberman
    Commented Nov 30, 2022 at 23:47
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    $\begingroup$ Do you have an example of an H-space map with finitely generated (non-trivial) kernel on homology, which doesn't just have $H_*(X)$ finitely generated? Feels weird to me, shouldn't there be some Hopf algebra magic that makes this impossible? $\endgroup$
    – Achim Krause
    Commented Dec 1, 2022 at 0:30

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