Let X be a K3 surface and $Y=X/\mathbb{Z}\_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi\_2(X)=\pi\_2(Y)$, while we know $H\_2(X)$ and $H\_2(Y)$ are very different. 


What are $\pi\_2(X)$ and $\pi\_2(Y)$?



Homotopy groups of K3 
Hurewicz theorem says that for a simply connected space $X$, $\pi\_2(X)\cong H\_2(X,\mathbb Z)$. So $\pi\_2(K3)\cong H\_2(K3,\mathbb Z)\cong \mathbb Z^{22}$. Here is a link:


<http://en.wikipedia.org/wiki/Hurewicz_theorem>


