I study braid groups and algebraic geometry, and I learned about the mapping class group (MCG).
Specifically, I encountered the MCG of the punctured unit disk (assume n punctures). I understand that the MCG of the punctured disk is a subgroup of the bijective diffeomorphisms group from the punctured disk to itself, and that diffeomorphisms are homotopic to the identity diffeomorphism on the boundary. How can I show that this subgroup is a normal subgroup of the bijective diffeomorphisms group?
If I name the diffeomorphisms group K, and the normal subgroup S, is the MCG K/S? If this is true, how do I prove this?