Realizing maps of braid groups by surface diffeomorphisms

Let Diff(D, zn) denote the group of smooth diffeomorphisms of the 2-dimensional disc that fix a neighborhood of ∂D and preserve a set zn consisting of n points. Let Diff0(D, zn) denote the identity component of this group. Then the mapping class group Diff(D, zn)/Diff0(D, zn) is isomorphic to Brn, the braid group on n strands. There is a natural “geometric” map ψ : Br2g+2 → Modg,2 induced by lifting mapping classes to a double cover Σg,2 of the disc D ramified over the points of z2g+2. One description of this map is as follows: Each f ∈ Diff(D, z2g+2) has a canonical lift to a homeomorphism of the cover Σg,2; this is the lift that fixes both boundary components pointwise. This gives an injective map Ψ : Diff(D, z2g+2)→ Homeo(Σg,2, ∂Σg,2), and the induced map on the quotient of these groups by their identity components is exactly ψ. Nariman [2] asks if these lifts can be made smooth: is there a map Diff(D, z2g+2)→ Diff(Σg,2, ∂Σg,2) that induces ψ on mapping class groups? Note that the construction above is inherently non-smooth: unless the derivative of f ∈ Diff(D, z2g+2) at each point z ∈ z is a scalar, the lift of f to a homeomorphism of the branched cover will not be differentiable at the branch points. Furthermore, there is some (weak) evidence to suggest that no “smoothing” is possible. For instance, Salter–Tshishiku [4] give obstructions to realizing braid groups by diffeomorphisms, so ψ cannot be obtained by a map that factors through Br2g+2. Work of Hurtado [1] also implies that such a map ψ should essentially be continuous, and that its restriction to the subgroup Diffc(D, z2g+2) of diffeomorphisms fixing a neighborhood of z (which we know to be nontrivial by [4]) must be obtained by embedding copies of covers of the open, punctured disc into Σg,2. This suggests, at least vaguely, that ψ would have to be obtained by branching the punctured disc over z, an inherently non-smooth construction. In [2], Nariman shows – perhaps surprisingly, given the above – that there is no cohomological obstruction to realizing ψ by a map on diffeomorphism groups. Here we confirm Nariman’s result and give an alternative proof, via an explicit construction of a realization.