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.



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> 1. 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 the *mapping class group* of $n$ points in $M$? Also, do I need any more restrictions on $M$? If so, why?
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* 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.



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> 2. Consider a presentation of $\text{Mot}\_{n}(M)$ with relations $R$.   
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> (i) Is it true that $\text{Mot}\_{n}(M)\cong B\_{n}(M)$, where $B\_{n}(M)$ is the surface braid group for $M$?  
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> (ii) Under what conditions will it be true that the generator relations $G$ of $B\_{n}$ will be a subset of $R$?
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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](https://arxiv.org/abs/1302.6536)


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).



Permuting $n$ points in a $2$-manifold
The comments seem to have answered questions 1 and 2i, to show that the group $\operatorname{Mot}\_n(M)$ is indeed the surface braid group $B\_n(M)$.


To answer 2ii, consider a disk $D \subset M$ such that $T \subset D$. Then the inclusion map $D \hookrightarrow M$ induces a homomorphism $B\_n \to B\_n(M)$, so the relations in $B\_n$ always hold in $B\_n(M)$.


