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?



Finite second homotopy group
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**](http://dx.doi.org/10.1016/j.crma.2010.01.011), *C. R. Math. Acad. Sci. Paris* **348** (2010), no. 5-6, 283--285.


and at the references given therein.


