Let $S\_{g,b}$ an orientable surface with genus $g$ and $b$ boundary components and $S\_g^b$ be an orientable surface with $b$ punctures. 


Denote by $PMCG(S\_g^b)$ and $PMCG(S\_{g,b}) $ the pure mapping class groups, that is, the group of orientation preserving homeomorphisms of the surface fixing the punctures or the boundary pointwise modulo isotopy. 


Is $$0 \to \mathbb Z^b \to PMCG(S\_{g,b}) \to PMCG(S\_g^b) \to 0 $$ a split short exact sequence ?



Mapping class groups of a punctured surface vs. surface with boundary
No, it is not split (except in a few degenerate cases like $(g,n) = (0,1)$ or $(g,n)=(0,2)$; let's assume that $g \geq 2$ for the moment just to be careful). It is a central extension, so if it was split then the abelianization of the pure mapping class group of $S\_{g,b}$ would contain a copy of $\mathbb{Z}^b$; however, this abelianization is trivial.


