TITLE: Higher-dimensional braid group?
QUESTION [6 upvotes]: Let $\Delta$ be 2-disk. Let $C(\Delta;n)$ be a configuration space. 
i.e.) $C(\Delta;n)= \lbrace (z_1,\ldots,z_n)\in \Delta\times\ldots\Delta | z_i\neq z_j ~\textrm{if}~ i\neq j \rbrace $
Then, it is well known or direct to see that $\pi_1(C(\Delta;n))= PB_n$, where $PB_n$ is original pure braid group of n-strands.
I heard that by using this configuration space we can easily generalize Braid group in arbitrary topological space $X$.
i.e.) We can define $PB_n(X)$ (Pure Braid group of n-strands in $X$) by $PB_n(X)=\pi_1(C(X;n))$. 
Also, we can study some exact sequences analogous to original braid group situation if topology of $X$ is good.
Here, but I heard that if $X$ is manifold with dimension greater than 2. Then, the structure of $PB_n(X)$ is somewhat trivial because there are enough rooms to move 
I think that this phenomena occurs because we are observing too small objects in large space. Roughly speaking, this seems to be the same situation that Knot theory is trivial in codimension greater than 2.
Instead of this formulation, I hope that we could modify or generalize this configuration space set up by observing some space containg information like distribution or foliation or something like that?
Can I do this business? In short, Can we think nontrivial generalization of Braid group in higher dimensional manifold?

REPLY [5 votes]: There are natural generalizations of the braid group and pure braid group to higher dimensions, as follows: consider the space of arrangements of $n$ (labelled) affine hyperplanes in ${\mathbb C}^k$ in general position. If $k=1$ this is the set of (labelled) configurations of $n$ distinct points in the plane, with fundamental group the classical n-strand (pure) braid group. The fundamental group of this space, for any $k\geq 1$, is not trivial, and can be considered a higher braid group - in fact I think it's sometimes been called that in the literature. 
The space of labelled generic arrangements can be described as the space of $({\mathbb C}^*)^n$-orbits of $n \times (k+1)$ complex matrices with all $(k+1) \times (k+1)$ minors nonzero, and is closely related to the realization space of the uniform matroid $U_{n,k+1}$, and the "generic stratum of the Grassmannian." The symmetric group $S_n$ acts freely on the space of labelled generic arrangements; the fundamental group of the space of $S_n$-orbits is the full higher braid group. 
The pure higher braid group plays a role in the Aomoto-Gelfand theory of generalized hypergeometric functions. I think it's fair to say that almost nothing, certainly very little, is known about the groups themselves. Personally, I'd love to see a nice presentation of the fundamental group of the space of unlabelled generic arrangements of $n$ lines in ${\mathbb C}^2$. It might not be too hard for $n=4$.