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I am interested in sufficient conditions for the second homotopy group $\pi_2(X)$ of a compact connected manifold to be finite. Are there familiar classes of manifolds $X$, for which this is the case?

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    $\begingroup$ Lie groups all have trivial $\pi_2$. $\endgroup$ Sep 6, 2016 at 12:30
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    $\begingroup$ Simply connected manifolds with finite $H_2(X;\mathbb{Z})$ $\endgroup$
    – Thomas Rot
    Sep 6, 2016 at 15:25
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    $\begingroup$ This is a condition on the universal cover, and so three large classes of manifolds with this property are the flat, hyperbolic, and (except when $n=2$) spherical manifolds, whose universal covers even all have trivial $\pi_2$. $\endgroup$ Sep 6, 2016 at 16:16
  • $\begingroup$ Are you aware that any finite dimensional connected CW-complex has the homotopy type of a compact connected manifold? Your question is too broad. $\endgroup$ Sep 6, 2016 at 20:13
  • $\begingroup$ Yes, the problem is broad. Therefore I am not asking for necessary and sufficient conditions and that is also why I am writing about "familiar classes of manifolds" which fulfill the condition. $\endgroup$ Sep 7, 2016 at 7:06

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You can have for instance a look at the following paper

Manuel Amann and Anand Dessai, MR 2600123 The $\hat A$-genus of $S^1$-manifolds with finite second homotopy group, C. R. Math. Acad. Sci. Paris 348 (2010), no. 5-6, 283--285.

and at the references given therein.

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