Braid Groups on Manifolds

Question

I am studying braid groups on manifolds and am getting confused. In a geometric definition, one needs to first choose a simple curve $\theta$ on a given manifold $M$ and well-ordered points $p_1,\cdots,p_n$ with respect to $\theta$. Intuitively it is clear; however, it seems to me that this definition depends on the choice of $\theta$ and $p_1,\cdots,p_n$. On the other hand, braid groups on manifolds can also be defined using the language of fundamental group of configuration spaces, without any simple curve $\theta$ mentioned. So my question is:

1. Does the braid group on a manifold depend on a choice of simple curve and well-ordered points on the curve?
2. What if the manifold $M$ is not path-connected? Does the braid group on $M$ still make sense in this case?

Answer 1

I will focus on the second part. If the manifold is not path connected, the definition of the braid group in terms of the fundamental group still makes perfect sense (although not so much in terms of the closed curve which passes through the points...). However, much like the fundamental group of any topological space, it will now strongly depend on which basepoint you choose!

For what follows, I will stick with the pure braid group, i.e. the fundamental group $\pi_1F(M,n)$.

Consider, for example, the space $X = D^2_x \sqcup D^2_y$, i.e. the disjoint union of two discs. Then the configuration space of two points in this is not path-connected, and it splits up into three components: two copies of $F(D^2_\alpha,2)$ and one copy of $D^2_x \times D^2_y$ (where the two points are in separate copies of $D^2$).

In this case, you have two possibilities, depending on components: if your basepoint is $(x_0, x_1) \in F(D^2_x,2)$ or is $(y_0, y_1) \in F(D^2_y,2)$, then you end up with $\mathbb{Z}$ as your braid group. If you instead choose $(x, y) \in D^2_x \times D^2_y$, then you end up with trivial fundamental group.

It isn't too hard to see that the configuration space of some disconnected space will decompose into components that are products of smaller configuration spaces. Hence the pure braid group will end up as a product of smaller braid groups on the components.

So at least for the pure braid group, you don't really get anything new when thinking about disconnected spaces.