TITLE: Why is a smooth algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ of degree 5 simply connected?
QUESTION [7 upvotes]: The title says it all.
In fact, i am only trying to prove that if $S$ is an irreducible smooth algebraic surface of degree 5 in $\mathbb{P}_{\mathbb{C}}^3$ (hence a four dimensional manifold over the reals), its first singular cohomology group $H_1(S ; \mathbb{Z})$ contains no two torsion.
I know only that by the short exact sequence of the sheaf of ideals and hodge theory that this group has no free part. But for the torsion part i have no idea.
Any help would be really appreciated, i would prefer hints over a complete answer!

REPLY [4 votes]: Hint: this follows from the Lefschetz Hyperplane Theorem using the Veronese (or "d-uple") Embedding.  See e.g. here for the details, if you like.