General relationship between braid groups and mapping class groups

Solution 1:

Let $\mathcal S$ be a compact surface, possibly with boundary. Let $\text{Homeo}^+(\mathcal S)$ refer to the group of orientation-preserving homeomorphisms that fix the boundary pointwise with the compact-open topology. Throwing an $n$ in there means we add $n$ marked points in the interior, which I prefer over deleting points. (I'd be worried about the validity of any particular version of this result if we delete points instead of marking them.) I don't know the correct topology on $\text{Homeo}^+$ if you use noncompact surfaces.

Then the correct general statement of your dream is that $$\text{Homeo}^+(\mathcal S,n) \to \text{Homeo}^+(\mathcal S) \to SF_n(\mathcal S)$$ is a fiber bundle. The tools you need to analyze this are the homotopy long exact sequence, the Earle-Eels theorem (you can find a statement and proof in Appendix B here, and that $\text{Homeo}^+(D^2)$ is contractible.

Some special cases:

When $\mathcal S = D^2$, you immediately obtain that $B_n = \text{MCG}(\mathcal S,n)$.

Ignore the issues with topologizing $\text{Homeo}^+(\mathbb R^2)$. The trick with your proof is that this is not actually simply connected! $\text{Homeo}^+(\mathbb R^2)$ should be the same as $\text{Homeo}^+(S^2,1)$ (send every homeomorphism to its compactification). This is homotopy equivalent to $SO(2) = S^1$.

When $\mathcal S = \Sigma_{g,k}$, a genus $g$ surface with $k$ boundary components, and either $g \geq 2$ or $k \geq 1$, applying Earle-Eels you obtain the exact sequence $$1 \to B_n(\mathcal S) \to \text{MCG}(\mathcal S,n) \to \text{MCG}(\mathcal S) \to 1.$$

When $\mathcal S = S^2$, using Smale's theorem that $\text{Homeo}^+(S^2) \simeq SO(3)$, and using Earle-Eels you can prove that the components of $\text{Homeo}^+(S^2,n)$ are contractible, so you get a short exact sequence $$1 \to \mathbb Z/2 \to B_n(S^2) \to \text{MCG}(S^2,n) \to 1.$$

When $\mathcal S = T^2$, $\text{Homeo}^+(T^2) \simeq SL_2(\mathbb Z) \times T^2$. If you can get some control on $\text{Homeo}^+(T^2,n)$ you should get something interesting here, but a couple mindless attempts didn't work. (Idea: work with Diff instead so that at each marked point you can 'pull apart' the marked points and obtain diffeomorphisms of $T^2$ minus some open discs without control on the way the diffeomorphisms behave on the boundary. This space might be assailable with EE.)

The fiber sequence above generalizes perfectly well to higher-dimensional manifolds. You might enjoy playing with it for 3-manifolds, when $\text{Homeo}^+(M)$ is known in many examples. $S^2 \times S^1$ might be fun.

You would probably enjoy Farb's primer on mapping class groups. The fiber sequence above just comes from modifying his proof of Birman's exact sequence to having $n$ points instead of one.