Research
Using Spin, Twist and Dial Homeomorphisms to Generate Homeotopy Groups
In this paper we consider some problems concerned with the isotopy classification of homeomorphisms of multiply punctured compact 2-manifolds, i.e., manifolds of the form X – ∪ m i=1 Definitions and Notation D̊ i – Pwhere X is a closed 2-manifold, {Di :1<i<m} is a family of disjoint discs in X and P = {pm+1, …, pn} is a finite subset of X disjoint from each Di. Inparticular we will show that various homeotopy groups for these manifolds are generated by the isotopy class of three types of homeomorphisms. In the case X is the two sphere we will give a complete presentation of these homeotopy groups. Parts of the material in this paper have been considered in [5 ], [6] and [7], but this is the first time that a full treatment of this topic, including detailed illustrations of the isotopies involved, has been submitted for publication. For an alternate approach to the material in this paper see (for example) [2], and [].
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 ðœ‹ðœ‹1(X,x0) denote the group of automorphisms of ðœ‹ðœ‹1(X,x0) where x0ðœ€ðœ€â€” Bd(Y). In this paper we establish the following result. Theorem. If Y is not aMoebius band or a disk, then H(Y)=Aut ðœ‹ðœ‹ 1 (X, x0) .
Boundary Fixed Homeomorphisms of 2-Manifolds with Boundary
Let X be a closed, orientable 2-manifold and let n X denote the bounded manifold obtained by removing the interiors of n disjoint closed disks from X .Let ( ) n H X denote the group of isotopy classes (rel boundary of n X ) of homeomorphisms of n X which are the identity on the boundary of n X . ( ) n H X has been determined for all n when X is the 2-sphere (see [8] and [10]). This paper investigates the structure of ( ) n H X for X not equal to the 2-sphere. In particular, a relationship between ( ) n H X and the homeotopy group (mapping class group)of X (see [4],[5]and [11]) is developed.
