$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\cd{cd}$Let $B_{k}(S_{g}),$ $\MCG(S_{g};k)$ and $\MCG(S_{g}))$ denote the braid group, the mapping class group (relative to $k$) and the mapping class group of the orientable surface $S_{g}$, respectively. For $g\geq3,$ we have a short exact sequence $$1\longrightarrow B_{k}(S_{g})\longrightarrow \MCG(S_{g};k)\longrightarrow \MCG(S_{g})\longrightarrow 1.$$ This short exact sequence gives the relation between the (virtual) cohomological dimensions: $$\vcd(\MCG(S_{g};k))\leq \cd(B_{k}(S_{g}))+\vcd(\MCG(S_{g})).$$ Does the same type of relation exist for punctured orientable surface $S_{g}-\{p_{1},\ldots,p_{n}\}?$
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