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Here is an interesting problem, which I could not solve and would appreciate any comment in solving it;

Assume that we have omitted infinitely many lines from $\mathbb{C}P^2$ to obtain $\mathbb{C}^2$ and now, consider the following lines in which $(z,w)$ is the coordinate of $\mathbb{C}^2$: $z=0, z=1, w=0, w=1, z=w.$

What is the fundamental group of the complement of these lines in $\mathbb{C}^2$?

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What you have is an arrangement of hyperplanes in $\mathbb C^2$ obtained by complexifying a real arrangement. There is a simple way to describe the fundamental group of its complement, due to Randell. You can find the details and references in the beautiful book Arrangement of hyperplanes by Peter Orlik and Hiroaki Terao, in section 5.3.

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Following up on JSE's answer, this arrangement is, up to linear change of coordinates, the projectivization of the braid arrangement of rank three, so the fundamental group of the complement is the quotient of the pure braid group (of the plane) on four strands by the infinite cyclic subgroup generated by the full twist (= the product of the usual generators), which is central. You can see this as follows: the four points are labelled x,y,z,w; the requirement that they are distinct means you throw away the hyperplanes x=y, x=z, x=w, etc. The complement of these six hyperplanes in C^4 is homotopy equivalent to the x=0 slice: the homotopy is given by sliding points parallel to the line x=y=z=w, which lies in all the hyperplanes. This gives the arrangement of six planes in C^3 with equations y=0,z=0,w=0,y=z,y=w,z=w. projectively, once you throw away y=0 you can set y=1 in the other five equations, and you get your arrangement.

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Up to PGL_2 action, by the way, this space is M_{0,5}, the space of 5 distinct points in P^1. (You've rigidified away the PGL_2 by sending the first three points to 0,1, infinity.) So the fundamental group ought to be some version of the spherical braid group on five strands (with some correction having to do with the PGL_2.) Maybe better is to think of your space as the configuration space of pairs of distinct points on CP^1 - 0,1,infty; in other words, you have the two-strand braid group on a pair of pants. Generators and relations for braid groups on punctured Riemann surfaces are well-known, I'm pretty sure -- I would look first at Farb and Margalit's book.

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    $\begingroup$ Randell's presentation is precisely a generalization of the usual presentation of the braid group: the braid arrangement is a complexified real arrangement. $\endgroup$ – Mariano Suárez-Álvarez Jun 20 '11 at 16:02

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