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?



Higher-dimensional braid group?
While I don't know about the **braid group**, there are certainly generalizations of **braids** to higher dimensions. In fact there is a huge literature on the subject. Perhaps the place to start is in Lee Rudolph's   
*Braided surfaces and Seifert ribbons for closed braids* Comment. Math. Helv. **59** (1983), 1-37.  

He defines a *braided surface* as follows. Let $D\_1\times D\_2$ be a product of 2-disks, and $\mathrm{pr}\_i:D\_1\times D\_2\rightarrow D\_i$ be the $i$th factor projection. Then a properly embedded, locally flat surface S in $D\_1\times D\_2$ is called a braided surface if:


* $\mathrm{pr}\_2|\_S:S\rightarrow D\_2$ is a branched covering.
* S is oriented so that $\mathrm{pr}\_2|\_S$ is orientation-preserving away from the (finite number of) branch points.


Rudolph proves that every ribbon surface in a 4-ball may be deformed into a braided surface. Braided surfaces are related to the (classical) braid group and to knot theory.  

Oleg Viro modified the above definition, and defined a *2-dimensional braid*, in order to obtain a concept which relates to higher-dimensional knot theory. Viro requires that $\partial S=P\_n\times \partial D\_2$, where $P\_n$ is a set of $n$ distinct interior points in $D\_1$. For this more restrictive definition, he proves that every surface link type is represented by a closure of a 2-dimensional braid. I don't know if Viro ever published his work, but [here is a reference](http://books.google.com/books?id=vjHK3VrcPvIC&pg=PA197&lpg=PA197&dq=surface+braids+topology&source=bl&ots=3Jn7E7ZLQ8&sig=MqEP1MzJJAVo_3NcbE3qoYNopas&hl=en&ei=j1F9TJmaFMmrnAe2oIidCw&sa=X&oi=book_result&ct=result&resnum=10&ved=0CEgQ6AEwCQ#v=onepage&q=surface%20braids%20topology&f=false).


