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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5$\begingroup$ Lie groups all have trivial $\pi_2$. $\endgroup$– Gabriel C. DrummondColeSep 6, 2016 at 12:30

4$\begingroup$ Simply connected manifolds with finite $H_2(X;\mathbb{Z})$ $\endgroup$– Thomas RotSep 6, 2016 at 15:25

4$\begingroup$ 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 $\pi_2$. $\endgroup$– Qiaochu YuanSep 6, 2016 at 16:16

$\begingroup$ Are you aware that any finite dimensional connected CWcomplex has the homotopy type of a compact connected manifold? Your question is too broad. $\endgroup$– Fernando MuroSep 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. $\endgroup$– William of BaskervilleSep 7, 2016 at 7:06
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1 Answer
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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. 56, 283285.
and at the references given therein.