In this book we will consider two fundamental objects attached to a surface S: a group and a space. We will study these two objects and how they relate to each other.
The group. The group is the mapping class group of S, denoted by Mod(S). It is defined to be the group of isotopy classes of orientation-preserving diffeomorphisms of S (that restrict to the identity on ∂S if ∂S ≠ ):
Mod(S) = Diff+(S, ∂S) / Diff0(S, ∂S).
Here Diff0(S, ∂S) is the subgroup of Diff+(S, ∂S) consisting of elements that are isotopic to the identity. We will study ...