Given $n$ points on a connected $2$-manifold $M$, I'd like to consider the homotopy classes of paths that "permute" these points.

**Edit** (Clarifying what I mean by this):

Given a set of $n$ distinct points $T=\{x_{1},\ldots,x_{n}\}\subset M$, to each point we assign a continuous simple curve $\gamma_{i}:[0,1]\to M$ such that $\gamma_{i}(0)=x_{i}, \gamma_{i}(1) \in T$ and $\gamma_{i}(s)\neq\gamma_{j}(s)$ for all $s\in[0,1]$ (where $i\neq j$). I'd like to consider the homotopy classes of all such possible curves.

- It seems obvious that these homotopy classes should constitute the elements of a group. Is that right? If so, what's the name of this group? I'm inclined to simply call this the
motion group$\text{Mot}_{n}(M)$ of the $n$ points on $M$. Does this coincide with themapping class groupof $n$ points in $M$? Also, do I need any more restrictions on $M$? If so, why?

E.g. considering $3$-space for a moment, it is obvious that $\text{Mot}_{n}(\mathbb{R}^{3})\cong S_{n}$ (where $S_{n}$ is the permutation group).

It is also obvious that $\text{Mot}_{n}(\mathbb{D}_{2})\cong \text{Mot}_{n}(\mathbb{R}^{2})\cong B_{n}$, where $\mathbb{D}_{2}$ is the $2$-disk with boundary and $B_{n}$ is the braid group.

- Consider a presentation of $\text{Mot}_{n}(M)$ with relations $R$.

(i) Is it true that $\text{Mot}_{n}(M)\cong B_{n}(M)$, where $B_{n}(M)$ is the surface braid group for $M$?

(ii) Under what conditions will it be true that the generator relations $G$ of $B_{n}$ will be a subset of $R$?

For instance, I'm sure that $\text{Mot}_{n}(S^{2})\cong B_{n}(S^{2})$ : in which case, we do have $G\subset R$ (in fact, $B_{n}(S^{2})$ is a quotient of $B_{n}$).

**Relevant Resources**:

A survey of surface braid groups and the lower algebraic K-theory of their group rings

I think my notion of $\text{Mot}_{n}(M)$ coincides with the Definition in Section 2.2 of the above paper. If so, then the answer to 2(i) is yes (according to the paper). Building on this, I believe Theorems 12 and 13 of Bellingeri (for $m=0$) may provide a partial answer to 2(ii).