Higher-dimensional braid group?

Question

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?

Answer 1

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.

Answer 2

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 $n=4$.

Answer 3

One possible generalization is the so-called "ring group" for $U_n$ - the fundamental group of the configuration space of unknots with $n$ components in the three-sphere. Brendle and Hatcher have a paper on this.

A generalization in a different direction is to consider the fundamental groups of the complements of complex hyperplane arrangements.

However, there cannot be any general theory about "fundamental groups of configuration spaces". (Consider configurations of a single point.)

Answer 4

As Sam Nead mentions, there are many generalizations of braid groups. Spaces of "unlinks" would be one of the most direct generalizations. Beyond the geometric aspect, there's a similar and somewhat more substantial connection to the automorphism group of a free group in this setting.

There is a large family of related ideas, one being embedding spaces. MO users such as Allen Hatcher, Dev Sinha, Tom Goodwillie, Pascal Lambrechts and myself (likely others too) have contributed to this area. Try going to the arXiv to see what they're up to. While you're at it, look up Victor Vassiliev's work and Victor Turchin/Tourtchine.

"this phenomena" you talk about, the main tool people use to prove such theorems is usually called transversality or general position.