Homeotopy Groups of 2aDimensional Manifolds with One Boundary Component
Let Y be a compact, connected 2—dimensional manifold with boundary. The homeotopy group of Y, denoted H(Y), is defined to be the group of isotopy classes in the space of all homeomorphisms of Yonto Y. This group (also known as the mapping class group) has been studied for various manifolds (see, for example, [2] and [3 ]). It is also possible to consider “subhomeotopy groups” where there are restrictions placed on the action of the homeomorphisms on the boundary of Y (see, for example, [7] and [8]). In this note we will consider the special case of a compact, connected manifold with exactly on boundary component. For the remainder of this paper we will assume Y represents a compact, connected manifold with exactly one boundary component and we willlet X denote the closed 2—manifold obtained by sewing a disk to the boundary of Y. Let Aut 𝜋𝜋1(X,x0) denote the group of automorphisms of 𝜋𝜋1(X,x0) where x0𝜀𝜀— Bd(Y). In this paper we establish the following result. Theorem. If Y is not aMoebius band or a disk, then H(Y)=Aut 𝜋𝜋 1 (X, x0) .
