Two-Dimensional Manifolds

Abstract

The term 'surface' is technically less specific but mostly used synonymous to '2-manifold' for which we will give a concrete definition. Topological 2-manifolds. Consider the open disk of points at distance less than one from the origin, D = {x ∈ R 2 | x < 1}. It is homeomorphic to R 2 , as for example established by the homeomorphism f : D → R 2 defined by f (x) = x/(1 − x). Indeed, every open disk is homeomorphic to the plane. Definition. A 2-manifold (without boundary) is a topological space M whose points all have open disks as neighborhoods. It is compact if every open cover has a finite subcover. Intuitively, this means that M looks locally like the plane everywhere. Exam-ples of non-compact 2-manifolds are R 2 itself and open subsets of R 2 . Examples of compact 2-manifolds are shown in Figure II.1, top row. We get 2-manifolds Figure II.1: Top from left to right: the sphere, S 2 , the torus, T 2 , the double torus, T 2 #T 2 . Bottom from left to right: the disk, the cylinder, the Möbius strip. with boundary by removing open disks from 2-manifolds with boundary. Alter-natively, we could require that each point has a neighborhood homeomorphic to either D or to half of D obtained by removing all points with negative first coordinate. The boundary of a 2-manifold with boundary consists of all points x of the latter type. Within the boundary, the neighborhood of every point x is an open interval, which is the defining property of a 1-manifold. There is only one type of compact 1-manifold, namely the circle. If M is compact, this implies that its boundary is a collection of circles. Examples of 2-manifolds