# K3 surface minus finite set

Let $S$ be a complex K3 surface, and $P\subset S$ a finite set of points in $S$. It is known that $$H^i(S,\mathbb{Z})\cong H^i(S\setminus P,\mathbb{Z})$$ for $0\le i \le 2$. Then the Euler characteristic computation implies that $b_3(S\setminus P)=|P|$. I want to confirm that $$H^3(S\setminus P,\mathbb{Z}) \cong \mathbb{Z}^{|P|},$$ that is, there is no torsion.

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What do you get from the Universal Coefficient Theorem? –  S. Carnahan Sep 8 '13 at 12:41
I don't think $P$ yields any torsion in $H_2(S\setminus P,\mathbb{Z})$, so the UCT implies that $H^3(S\setminus P,\mathbb{Z})\cong Hom(H_3(S\setminus P,\mathbb{Z}),\mathbb{Z})$, but does this help? –  Sohrab Sep 8 '13 at 12:53
I think it does, since $\mathrm{Hom}(T,\mathbb Z)$ vanishes if $T$ is torsion... –  Dan Petersen Sep 8 '13 at 14:22
Sohrab, I think your Euler characteristic is off by one. Using Poincaré-Lefschetz duality will help give you a clean answer. –  Tim Perutz Sep 8 '13 at 14:46