Let $MCG(S_{g,b}^s)$ be the mapping class group of a surface $S_{g,b}^s$.
Is it true that $MCG(S_{g,b}^s)$ is isomorphic to $MCG(S_{g',b'}^{s'})$ if and only if $ S_{g,b}^s$ is homeomorphic to $S_{g',b'}^{s'}$ ?
Isomorphism type of mapping class group
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