4
$\begingroup$

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 $\pi_2(Y)$?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
8
$\begingroup$

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

Improve this answer
$\endgroup$
2
  • $\begingroup$ Don't you mean 22? $\endgroup$
    – Will Sawin
    Commented Jan 28, 2013 at 19:11
  • 1
    $\begingroup$ Sure Will :) that was a misprint $\endgroup$
    – Dmitri Panov
    Commented Jan 28, 2013 at 19:12

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .