DOC PREVIEW
Duke CPS 296.1 - Two-dimensional Manifolds

This preview shows page 1-2 out of 6 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 6 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 6 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 6 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

26 II SurfacesII.1 Two-dimensional ManifoldsThe 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 lessthan one from the origin, D = {x ∈ R2| kxk < 1}. It is homeomorphic toR2, as for example established by the homeomorphism f : D → R2defined byf(x) = x/(1 − kxk). Indeed, every open disk is homeomorphic to the plane.Definition. A 2-manifold (without boundary) is a topological space Mwhose points all have open disks as neighborhoods. It is compact if everyopen cover has a finite subcover.Intuitively, this means that M looks locally like the plane everywhere. Exam-ples of non-compact 2-manifolds are R2itself and open subsets of R2. Examplesof compact 2-manifolds are shown in Figure II.1, top row. We get 2-manifoldsFigure II.1: Top from left to right: the sphere, S2, the torus, T2, the double torus,T2#T2. Bottom from left to right: the disk, the cylinder, the M¨obius strip.with boundary by removing open disks from 2-manifolds with boundary. Alter-natively, we could require that each point has a neighborhood homeomorphicto either D or to half of D obtained by removing all points with negative firstcoordinate. The boundary of a 2-manifold with boundary consists of all pointsx of the latter type. Within the boundary, the neighborhood of every point xis an open interval, which is the defining property of a 1-manifold. There isonly one type of compact 1-manifold, namely the circle. If M is compact, thisimplies that its boundary is a collection of circles. Examples of 2-manifoldsII.1 Two-dimensional Manifolds 27with boundary are the (closed) disk, the cylinder, and the M¨obius strip, allillustrated in Figure II.1, bottom row.We get new 2-manifolds from old ones by gluing them to each other. Specifi-cally, remove an open disk each from two 2-manifolds, M and N, find a homeo-morphism between the two boundary circles, and identify corresponding points.The result is the connected sum of the two manifolds, denoted as M#N. Form-ing the connected sum with the sphere does not change the manifold since itjust means replacing one disk by another. Adding the torus is the same asattaching the cylinder at both boundary circles after removing two open disks.Since this is like adding a handle we will sometimes refer to the torus as thesphere with one handle, the double torus as the sphere with two handles, etc.Orientability. Of the examples we have seen so far, the M¨obius strip hasthe curious property that it seems to have two sides locally at every interiorpoint but there is only one side globally. To express this property intrinsically,without reference to the embedding in R3, we consider a small, oriented circleinside the strip. We move it around without altering its orientation, like a clockwhose fingers keep turning in the same direction. However, if we slide the clockonce around the strip its orientation is the reverse of what it used to be and wecall the path of its center an orientation-reversing closed curve. There are alsoFigure II.2: Left: the projective plane, P2, obtained by gluing a disk to a M¨obiusstrip. Right: the Klein bottle, K2, obtained by gluing two M¨obius strips together.The vertical lines are self-intersections that are forced by placing the 2-manifolds inR3. They are topologically not important.orientation-preserving closed curves in the M¨obius strip, such as the one thatgoes around the strip twice. If all closed curves in a 2-manifold are orientation-preserving then the 2-manifold is orientable, else it is non-orientable.Note that the boundary of the M¨obius strip is a single circle. We can therefore28 II Surfacesglue the strip to a sphere or a torus after removing an open disk from the latter.This operation is often referred to as adding a cross-cap. In the first case we getthe projective plane, the sphere with one cross-cap, and in the second case weget the Klein bottle, the sphere with two cross-caps. Both cannot be embeddedin R3, so we have to draw them with self-intersections, but these should beignored when we think about these surfaces.Classification. As it turns out, we have seen examples of each major kindof compact 2-manifold. They have been completely classified about a centuryago by cutting and gluing to arrive at a unique representation for each type.This representation is a convex polygon whose edges are glued in pairs, called apolygonal schema. Figure II.3 shows that the sphere, the torus, the projectiveplane, and the Klein bottle can all be constructed from the square. Moreb abb baba abb ba a a aaaaaa a a ab bbb aa aaFigure II.3: Top from left to right: the sphere, the torus, the projective plane, andthe Klein bottle. After removing the (darker) M¨obius strip from the last two, we areleft with a disk in the case of the projective plane and another M¨obius strip in thecase of the Klein bottle. Bottom: the polygonal schema in standard form for thedouble torus on the left and the double Klein bottle on the right.generally, we have a 4g-gon for a sphere with g handles and a 2g-gon for asphere with g cross-caps attached to it. The gluing pattern is shown in thesecond row of Figure II.3. Note that the square of the torus is in standardform but that of the Klein bottle is not.II.1 Two-dimensional Manifolds 29Classification Theorem. The two infinite families S2, T2, T2#T2, . . . andP2, P2, P2#P2, . . . exhaust the family of compact 2-manifolds without boundary.To get a classification of compact 2-manifolds with boundary we can take onewithout boundary and make h holes by removing the same number of opendisks. Each starting 2-manifold and each h ≥ 1 give a different surface andthey exhaust all possibilities.Triangulations. To triangulate a 2-manifold we decompose it into triangu-lar regions, each a disk whose boundary circle is cut at three points into threepaths. We may think of the region and its boundary as the homeomorphic im-age of a triangle. By taking a geometric triangle for each region and arrangingthem so they share vertices and edges the same way as the regions we obtaina piecewise linear model which is a triangulation if it is homeomorphic to the2-manifold. See Figure II.4 for a triangulation of the sphere. The conditionFigure II.4: The sphere is homeomorphic to the surface of an octahedron, which is atriangulation of the sphere.of homeomorphism requires that any two triangles are


View Full Document

Duke CPS 296.1 - Two-dimensional Manifolds

Documents in this Course
Lecture

Lecture

18 pages

Lecture

Lecture

6 pages

Lecture

Lecture

13 pages

Lecture

Lecture

5 pages

Load more
Download Two-dimensional Manifolds
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Two-dimensional Manifolds and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Two-dimensional Manifolds 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?