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?