So, after the last topology post, we know what a manifold is – it’s a structure where the neighborhoods of points are *locally* homeomorphic to open spheres in some ℜn.
We also talked a bit about the idea of *gluing*, which I’ll talk about
more today. Any manifold can be formed by *gluing together* subsets of ℜn. But what does *gluing together* mean?
Let’s start with a very common example. The surface of a sphere is a simple manifold. We can build it by gluing together *two* circles from ℜ2 (a plane). We can think of that as taking each circle, and stretching it over a bowl until it’s shaped like a hemisphere. Then we glue the two hemispheres together so that the *boundary* points of the hemispheres overlap.
Now, how can we say that formally?
Well, first we need to define what a sphere is in non-topological terms.
The surface of a three-dimensional sphere with radius N is the set of points (x,y,z) that satisfy the equation x2 + y2 + z2 = N2. For the pedantic types among us, the sphere with radius N is the set {(x,y,z) ∈ ℜ3 : x2 + y2 + z2 = N2}.
Then, since the surface of a sphere is locally two-dimensional (that is, it’s a 2-manifold) we need to define how to map from subsets of ℜ2 to the surface of the sphere. For it to work, it needs to be an *invertible* function – that is, a function *f : X → Y* where both *f* and *f-1* are functions, and *∀ x : f-1(f(x))=x*.
For a sphere that’s easy. We’ll start by describing the halves. For one half, we define a mapping from the hemisphere to a disk in ℜ2 as a function f1 : ℜ3 → ℜ2. For each point (x,y,z) in the sphere where z≥0, we can map it onto a circle using: f1(x,y,z) = (x,y). For the second half, we’ll define f2(x,y,z) = (x,y), but this time we’ll say that f2 is only defined for points where z≤0.
And now, for the gluing. Gluing is done using a special function called a *transition map*. A transition map for a pair of *n*-manifolds A and B is a pair of invertible functions *tA, tB*, where tA maps from A to an open-ball in ℜN, and tB does the same for B. The manifolds are *glued* by the transition map if for every point *x* in the overlap between A and B, there is an invertible mapping from *A* to *B* provided by *tB-1(tA(x))*, and vice versa.
The transition map for the sphere is very simple. The overlap is the *boundaries* of the two circles – the very outer perimeter, the region where z=0. The transition map is *exactly* the same functions that we used to map the two circles onto the sphere.
So using that mapping, we can have a clean “glue line” between the two – there’s exactly one point of overlap, and it’s the circle where the sphere meets the XY plane. And we have a definition of the sphere as a manifold that gives us a metric – the *coordinates* on the circles can be used to describe locations on the sphere.
I like these topology posts.
For “gluing,” does the formal definition always stay within our intuitive understanding of the concept? Are there any cases of gluing that follow mathematically, but don’t seem like “gluing” from an intuitive standpoint?
I ask because I want to make sure I fully understand the definition.
Off the top of my head, they do mostly seem like gluing. There are two catches:
1) The intuitive idea of glue is really based on metric properties of the manifold – but not all manifolds are metric.
2) A lot of our intuitions about gluing are based on two and three dimensional constructs – but the topological spaces being glued can end up with many more dimensions.
There’s a big problem with this definition, in that properly the gluing should occur on an overlap of the patches. Also, what you’ve called the “transition map” is really the inverse of the coordinate map for the patch. The transition map goes from part of one coordinate patch to another.
Say we take the same two disks, but stretch them a little past the equator. Then the coordinate maps send a point on the sphere in the image of one disk to the corresponding point in that disk, and a point in the image of the other disk into the corresponding point in that disk. The transition map sends an annulus at the edge of one disk onto an annulus at the edge of the other.
The whole point of the transition map is that each disk is a part of a real plane, and so each has a notion of which functions on it are differentiable. A function defined in one coordinate patch on the sphere can be called differentiable if it’s differentiable as a function of the coordinates of the patch. If the transition function from a part of one patch to a part of the other is itself differentiable, then it doesn’t matter which patch we use to determine if a function is differentiable at a point — it’s either differentiable in all coordinate patches containing that point or it’s differentiable in none of them. And to determine if the transition function transforming one coordinate patch to another is differentiable it needs to be defined on an open set on the manifold. The overlap cannot be just a line like you indicate.
@John:
What you’re saying is true in general, but it’s okay to glue two manifolds-with-boundary together along a common boundary, as long as you’re careful. Each boundary point has a neighborhood which looks like the closed half-space, and the glued result looks like R^n.
@everybody:
That said, the treatments I’ve seen give either two or 2(n+1) charts for the n-sphere, all with open overlaps, as John said. Also it’s often better to have open overlaps for technical reasons, like imposing a metric, or looking at changes of variables.
Your discussion would probably be clearer if you didn’t use “circle” to mean both “circle” and “disk”, and if you didn’t use “sphere” to mean both “sphere” and “ball”.