Chapter Two Mapping Class Group Basics In this chapter we begin our study of the mapping class group of a surface. After giving the definition, we compute the mapping class group in essen- tially all of the cases where it can be computed directly. This includes the case of the disk, the annulus, the torus, and the pair of pants. An important method, which we call the Alexander method, emerges as a tool for such computations. It answers the fundamental question: how can one prove that a homeomorphism is or is not homotopically trivial? Equivalently, how can one decide when two homeomorphisms are homotopic or not? 2.1 DEFINITION AND FIRST EXAMPLES Let S be a surface. As in Chapter 1, we assume that S is the connect sum of g 0 tori with b 0 disjoint open disks removed and n 0 points removed from the interior. Let Homeo + (S, S) denote the group of orientation-preserving homeomorphisms of S that restrict to the identity on S. We endow this group with the compact-open topology. The mapping class group of S, denoted Mod(S), is the group Mod(S) = 0 (Homeo + (S, S)). In other words, Mod(S) is the group of isotopy classes of elements of Homeo + (S, S), where isotopies are required to fix the boundary point- wise. If Homeo 0 (S, S) denotes the connected component of the identity in Homeo + (S, S), then we can equivalently write Mod(S) = Homeo + (S, S)/ Homeo 0 (S, S). The mapping class group was first studied by Dehn. He gave a lecture on this topic to the Breslau Mathematics Colloquium on February 22, 1922; see [49]. The notes from this lecture have been translated to English by Stillwell [51, Chapter 7]. There are several possible variations in the definition of Mod(S). For ex- ample, we could consider diffeomorphisms instead of homeomorphisms, or homotopy classes instead of isotopy classes. By the theorems in Section 1.4,