For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then simplified in [2].

How about in dimension $3$? The first question is that, unlike in dimension $2$ in which the classification of spaces is easy (genus and number of boundaries), in dimension $3$ the classification is complicated. One either uses

- Thurston's geometrization
- Lickorish and Kirby's presentation using links

Neither of them is easy, so I'd expect it much harder to get presentation for the mapping class group.

**Question**: Nevertheless, is there any presentation known?

In the first view-point, the closest answer I have seen is this, in which Allen Hatcher claims that the MCG of three-manifolds are essentially known [3]. There, the natural map from MCG to $Out(\pi_1(M))$ is considered. While the kernel of this map is understood in the nonprime case [3. section 2], it is not necessarily onto. Even if it's onto, we still do not have a presentation for MCG.

In the second view-point, I have not heard of any result.

### Reference

[1] A presentation for the mapping class group of a closed orientable surface-[Hatcher and Thurston]

[2] A simple presentation for the mapping class group of an orientable surface-[Bronislaw Wajnryb]

[3] Stabilization for mapping class groups of 3-manifolds-[Allen Hatcher and Nathalie wahl]

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