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We show that the group cohomology of the diffeomorphisms of the disk with $n$punctures has the cohomology of the braid group of $n$ strands as the summand.As an application of this method, we also prove that there is no cohomologicalobstruction to lifting the "standard" embedding$\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphismbetween diffeomorphism groups.

Braid groups and discrete diffeomorphisms of the punctured disk