Homotopy groups of K3 - MathOverflow most recent 30 from http://mathoverflow.net2013-06-19T18:48:28Zhttp://mathoverflow.net/feeds/question/120139http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/120139/homotopy-groups-of-k3Homotopy groups of K3 Mohammad F.Tehrani2013-01-28T18:56:46Z2013-01-28T19:12:10Z
<p>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. </p>
<p>What are $\pi_2(X)$ and $\pi_2(Y)$?</p>
http://mathoverflow.net/questions/120139/homotopy-groups-of-k3/120141#120141Answer by Dmitri for Homotopy groups of K3 Dmitri2013-01-28T19:06:59Z2013-01-28T19:12:10Z<p>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:</p>
<p><a href="http://en.wikipedia.org/wiki/Hurewicz_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Hurewicz_theorem</a></p>