Homotopy groups of K3 - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:48:28Z http://mathoverflow.net/feeds/question/120139 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120139/homotopy-groups-of-k3 Homotopy groups of K3 Mohammad F.Tehrani 2013-01-28T18:56:46Z 2013-01-28T19:12:10Z <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#120141 Answer by Dmitri for Homotopy groups of K3 Dmitri 2013-01-28T19:06:59Z 2013-01-28T19:12:10Z <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>