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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 $\pi_2(Y)$?

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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

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  • $\begingroup$ Don't you mean 22? $\endgroup$ – Will Sawin Jan 28 '13 at 19:11
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    $\begingroup$ Sure Will :) that was a misprint $\endgroup$ – Dmitri Panov Jan 28 '13 at 19:12

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