Euler Characteristic and Genus

It turns out that the Euler characteristic, together with the number of boundary components, completely determines a surface. We can’t prove this, but we can discuss it and related ideas!

So surfaces split up into groups in a number of ways. Here are a two: Boundary vs. no boundary Some surfaces we have encountered, like a Mobius strip or a single 2-cell have an edge. We can identify an edge by looking for 1-cells that are attached on only one side—there aren’t two 2-cells attached to them (which might be the same 2-cell attached twice!). We don’t like these, because these edges, which are called boundary components are different than the rest of the surface. We’d really like to study things that are homogeneous, and points along the boundary don’t work like points elsewhere. Now some surfaces (e.g., a plane, an infinite cylinder) don’t have a boundary because any boundary would be infinitely far away. But other, more “finite-looking” surfaces might not have a boundary, either. Consider a cylinder where we leave the edges “fuzzy.” That is, they are open sets instead of closed sets that contain their boundary components. In terms of continuity, we could stretch these pieces out to infinity and they would be the same surface. The way to think about it is looking at the hyperbolic plane. As you approach the x-axis, distances get larger and larger, and you can’t really get to the x-axis. So the plane extends to infinity without really having to be stretched out that far. We can always do this with any boundary component of a surface, so infinite surfaces and bounded but boundary-less surfaces really are the same thing as far as we’re concerned.

Infinite vs. finite As we’ve just discussed, infinite surfaces can be thought of as boundaryless finite surfaces, and the topology is the same. Certainly, they can be split into cells the same way! And we don’t want to deal with the infinite, so we’ll just say all surfaces should be made finite and boundary-less if necessary.

So our idea surfaces are finite without boundaries. We can fix infinite surfaces, but then they have boundaries. How do we fix that? Well, all boundaries are loops. Simply attach a 2-cell around this loop! This adds one to the number of 2-cells, but removes a boundary component.

Theorem Closed surfaces are determined, upto deformations, by their Euler characteristic and their orientability.

Corollary Finite surfaces with boundaries are determined, upto deformations, by their Euler characteristic, orientability, and the number of boundary components.

We won’t prove these; it’s beyond our ability at present. But let’s see what we get from them.

1 The simplest surface is a 2-cell attached to a single 0-cell—all the boundary is collapsed down to s single point. This results basically in a sphere. The Euler characteristic is 1 − 0 + 1 = 2. Note that every cell structure for a sphere gives an Euler characeristic of 2, because every the most basic one does, and any other can be built up from this one by applying the procedures from last time, which do not change the Euler characteristic. What does each boundary component do to the Euler characteristic? Well, since we could plug the hole by sewing in a 2-cell patch, which would increase the characteristic by one, we know that every boundary component decreases the characteristic by one. What about our doubled-up 2-cell? Now we have one each of a 2-cell, a 1-cell, and a 0-cell. The characteristic is one. All cell structures on the projective plane will give this same Euler characteristic. Let’s use the standard notation χ to represent the Euler characteristic. Let’s try a Mobius strip. We found a cell structure with two 2-cells, six 1-cells, and four 0-cells, so χ = 2 − 6 + 4 = 0. Let’s fill in the boundary. Since the boundary is a single loop, we attach to it a single disk, though this can’t be done inside of three-dimensional space without the figure crossing through itself. As we know, filling in a hole like this adds a 2-cell, so increases χ by one. Thus, the resulting closed surface has χ = 1. That means it must be a projective plane! On the other hand, what would happen if we glued two Mobius strips together along their boundary? The resulting shape would now have four 2-cells, eight 1-cells, and four 0-cells, for a χ of zero. This closed, non-orientable (1-sided) surface is therefore different than the projectve plane. It is called a Klein bottle. If we stick to two-sided surfaces, they can all stay inside three dimensions. They might be all knotted up, but it turns out that all we will care about is the number of “handles.” Here is a rule. Let’s say you have two surfaces, and you glue them together. Or even gluing together two separate parts of a single surface. This can be accomplished by cutting a number n of disks out of each, and attaching the two surfaces together along these cuts. Mathematical mad scientist surgery! (The process is actually called surgery!) If we attach two surfaces together along just one disk, it is called the connected sum of the two surfaces. Step-by-step, what does this do to χ? Drawing the disks that will be cut out requires adding n 1-cells and n 0-cells to draw lines out to where the disk will be cut out (the first step below). This doesn’t change χ of course. Then we draw the disks, which creates n new 1-cells and n 2-cells. This doesn’t change χ either.

Now we cut out all the disks—decreasing the characteristic of each surface by n. Then, when we glue the surfaces together, n 1-cells and n 0-cells disappear as the corresponding piece of each hole is glued to the hole in the second surface. The overall effect is that we have decreased the total χ by 2n.

2 It turns out we can create any surface by doing this kind of surgery, which has the effect of adding the two characteristics for the surfaces and then subtracting twice the number of places where we’re attaching them together. It is actually more common to glue parts of a surface to itself. Doing so once is called “attaching a handle.” Each time we attach a handle, we decrease χ by two. The number of handles a surface has is called it genus, and we have now explained the formula (for orientable surfaces) χ = 2 − 2g. So start with a sphere, χ = 2. Now attach a handle, and you have a torus. According to our work, χ = 0 for this torus. Is this right? Sure! A torus can be created by taking a single 2-cell, gluing the top to the bottom, and then gluing the sides together. This will cut the number of 1-cells to two, and glues all 0-cells together. So F − E + V = 1 − 2 + 1 = 0. Handles are a way from getting from one part of a surface to another without actually going along the usual route. So maybe that extra route will allow us to solve the WGE problem? On a torus, we still have nine edges and six vertices, but now if there is a solution it only includes three faces. With nine edges, that means 18 total sides to our faces, so if split evenly each face will have six sides, and thus contain all six vertices. And, lo, it turns out we can build it! Can WGE be solved on a Mobius strip? How about a projective plane? (Hint: if it can be solved on the Mobius strip, it can on the projective plane, since the latter is the former with an extra 2-cell sewn in!) Another application: can the complete graph on five vertices be embedded in the plane? That is, if we have five points and connect each point to each other, can this be drawn in the plane? No! For this would create a cell structure with five 0-cells and ten 1-cells (5C2 = 10), and thus a necessity for seven 2-cells. Each 2-cell will have at least three sides, leading to at least 21 sides total. Since each side belongs to two 2-cells, we get at least 10.5 1-cells, but we only have ten! Can this shape be embedded in a torus? A Mobius strip?

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