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I guess that the following are true; maybe classical? Is there a reference?

  1. Let $X, Y, Z$ be connected pointed CW-complexes, $f:X\to Y$ and $g:X\to Z$ pointed maps. Assume that for every $k\ge 1$, the kernel of $f_*:\pi_k(X)\to\pi_k(Y)$ is contained in the kernel of $g_*:\pi_k(X)\to\pi_k(Z)$. Does it follow that there is a pointed map $h:Y\to Z$ such that $h\circ f$ is homotopic to $g$?

  2. Let $\mathrm{Diff}(M)$ be the group of the smooth diffeomorphisms of a closed separated manifold $M$, with the smooth topology. Does $\mathrm{Diff}(M)$ have the homotopy type of a numerable CW-complex? Are its integral homology groups numerable?

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    $\begingroup$ (1) is very, very false. In general the problem of factorizing a map through another up to homotopy is attacked via obstruction theory and it's not trivial. $\endgroup$
    – Denis Nardin
    Commented Mar 27, 2021 at 22:28
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    $\begingroup$ No to 1- : take $X=Y=Z = S^1$, $f$ any index $2$ map and $g$ the identity (and here the obstruction is already at the group-theoretic level ! there are examples with "no group theoretic" obstructions but subtler homotopy theoretic obstructions) $\endgroup$
    – Maxime Ramzi
    Commented Mar 27, 2021 at 22:28
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    $\begingroup$ (2) As $\mathrm{Diff}(M)$ is a Frechet manifold, it is homotopy equivalent to a countable CW-complex and hence has countable homology groups. Look at the work of Michor. $\endgroup$
    – skupers
    Commented Mar 27, 2021 at 22:57
  • $\begingroup$ Thank you, Denis and Maxime, for the pertinent answers (to a stupid question). Thank you, Skupers, for the reference to an author whom I should have known. $\endgroup$
    – Gael Meigniez
    Commented Mar 28, 2021 at 10:22
  • $\begingroup$ By the way, mathoverflow.net/a/126841/798 gives detailed references for (2). $\endgroup$
    – skupers
    Commented Mar 28, 2021 at 15:57

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