Constructing new spaces out of old - Connected Sums

In this section we will consider a standard topological construction,which will allow us to construct new surfaces. This construction is a particular case of a more general one which will be introduced in the following chapter.

We start by an easy example of a much more general technique. Namely let S and T be two surfaces. By definition every point in the interior of a surface has a neighbourhood homeomorphic to an open disk. Thus given S and T it is possible to remove an open disk from S and another from T and then glue the resulting spaces along the circular boundary components thus created. This operation is called the connected sum of S and T and is denoted S#T.

Although the construction is intuitively clear, it is worthwhile making the definition of connected sums a bit more rigorous. This is done as follows: Let T and S be surfaces as before. Choose any two interior points points x_S $\in$ S and x_T $\in$ T and let D_S and D_T denote open disks in S and T respectively, which are contained entirely in the interior of the respective surfaces. Such disks can clearly be chosen since x_S and x_T were taken to be interior points. Let $\bar{S}$ and $\bar{T}$ denote the surfaces (with boundary) obtained by removing the disks D_S and D_T. Let f_S : S¹$\rTo$$\bar{S}$ and f_T : S¹$\rTo$$\bar{T}$ be any two choices of embeddings of the circle into the boundaries $\partial$D_S and $\partial$D_T respectively. Now, let S#T denote the surface obtained from $\bar{S}$ and $\bar{T}$ by identifying any point y' $\in$ $\partial$D_S with the point y'' $\in$ $\partial$D_T if there exists some point z $\in$ S¹ such that f_S(z) = y' and f_T(z) = y''.

Exercise 3.1   Prove that the operation of taking connected sums is well defined up to homeomorphism, namely that it does not depend on

1.

which disks are removed from the respective surfaces as long as both are open and contained entirely in the interior and

2.

the choice of the homeomorphisms f_S and f_T.

Connected sums allow us to create a larger variety of surfaces by taking an iterated connected sum of surfaces we are already familiar with. For instance, we can create surfaces with as many ``holes'' as we want by taking an iterated connected sum of tori, or alternatively attaching handles to a sphere. In fact connected sums enables us to construct all surfaces out of 2-sphere, the cylinder, and the Möbius band or alternatively, replace the cylinder by the torus.

We can now start a more rigorous discussion of surfaces. The basic examples or closed surfaces we have so far are

1.

the 2-sphere and the torus (closed orientable surfaces) and

2.

the projective plane and the Klein Bottle (closed, non-orientable surfaces).

In addition we have a small variety of surfaces with boundary. An important example to keep in mind of a non-orientable surface with a boundary is the Möbius band. The aims of this section are

1.

To discuss more complicated surfaces, which one can get by taking connected sums of the surfaces already introduced

2.

To introduce the concept of embedding of a surface in Euclidean space and demonstrate that homeomorphic surfaces can appear in different guises, depending on their embeddings

3.

To discuss a classification theorem, which at this point we will not be able to prove, characterising all closed surfaces up to homeomorphism and a choice of embedding.

Consider the 2-sphere S². One can cut off two open disks out of the sphere and connect a handle, i.e. a cylinder with two circular boundary components to the holes thus created. The resulting space is easily observed to be homeomorphic to the torus. Alternatively, since we are already comfortable with the torus itself, observe that the same result is obtained by taking the connected sum of the torus and the 2-sphere, although doing that just to obtain the torus again might look a bit pointless.

However, there is no reason in both constructions, why we should limit ourselves to carrying it out only once. One can indeed cut off any even number 2n of holes in a sphere and connect handles to create a sphere with n-handles. On the other hand it is also possible to use connected sums to cut and clue together any number of tori. Two questions arise immediately.

1.

Is a sphere with n handles homeomorphic to a connected sum of n-tori?

2.

Once more than one handle is attached to a 2-sphere, there are many possible way to do the attachment. Namely, the handles can be knotted within themselves in highly non-trivial ways. Does the number of handles uniquely determine the topological type of the resulting surface, or do we have to distinguish between surfaces if the handles are knotted in different ways.

If the handles are attached to the sphere so that they are not knotted it is easy to see that a sphere with n handles is homeomorphic to a connected sum of n tori. To do this just imagine that you have the sphere with n handles in your hands and by pulling and stretching it you can bring it to the desired shape without breaking it. This gives an intuitive positive answer to the first question.

The second is a bit more tricky. Namely, it is clear that just by pulling and stretching you can't bring a knotted sphere with handles to the shape of a torus. However, remember that when you visualise this geometric object you are really thinking about it as a subspace of Euclidean 3-space. Clearly the dimension of the space in which the object is embedded does not effect its homeomorphism type. Thus try to imagine the handled sphere as a subspace of R⁴. Just as the Klein bottle can be constructed in 4-space without tearing the surface, you can shift handles around in our case to bring the sphere with n handles to the required form of a connected sum of n tori. This shows that knotting of the handles makes no difference to the homeomorphism type of the surface.

The question now becomes, what is the feature of a connected sum of n tori which could distinguish it from a connected sum of n + 1 tori. You could try to answer an instance of this question in the following.

Exercise 3.2   Show that a 2-sphere is not homeomorphic to the torus.

So far we have been discussing orientable closed surfaces. Namely, observe that a sphere with any number of handles has a well defined ``inside'' and ``outside''. This is far from being a rigorous definition of orientability, but it will do for now. One can of course consider connected sums of arbitrary surfaces including orientable and non-orientable put together. A fundamental construction corresponding to attaching handles to spheres is removing n disjoint open disks from a sphere and replacing each one of them by a Möbius band. Remember that a Möbius band has only ``one side'' and in particular its boundary consists of a single circle. Thus this construction makes sense, as long as one thinks about it as occuring in R⁴. Examples of the surfaces obtained this way were already given. Namely, for n = 1 one gets the projective plane and n = 2 gives the Klein Bottle.

To complicate things even more, let us now discuss in some detail the concept of embeddings.

Definition 3.3   Let f : X$\rTo$Y be a continuous map. Then f is called an embedding if, considered as a map from X to the image subspace Im(f ) $\subseteq$ Y, f is a homeomorphism.

Two subspaces V and W of the same space Y may be just two different embeddings of the same space, say X. As topological spaces V and W are the same space but for some applications one might like to consider the embedding as part of the data. A particularly nice example was already mentioned. Namely, one can embed a sphere with n handles in R³ in a garden variety of ways, knotting the handles to one's heart content. Nevertheless, the only difference among all those possible embeddings is the embedding itself and not the space under consideration. How does one distinguish two surfaces then? The answer to that will come later when we consider some topological invariants one can use to solve this problem.

We now state a beautiful theorem, without proof at this stage, which gives a complete classification of all surfaces. To the reader whose imagination has been fired by what appears to be a fantastic variety of different surfaces one can construct, this theorem should be quite surprising.

Theorem 3.4   Any closed surface is homeomorphic either to a sphere, or to a sphere with a finite number of handles attached to it, or to a sphere with a finite numbers of discs removed and replaced by Möbius bands. Up to homeomorphism every surface arises this way and no two among these are homeomorphic.

A curious point about this theorem appears to be the assertion that mixed types are not allowed in the classification. Indeed it can be showed for example that a sphere with one handle where a small disk was removed and replaced by a Möbius band is homeomorphic to a sphere with three disjoint disks removed and replaced by Möbius bands.

A sphere with n handles attached is called a closed orientable surface of genus n. The genus, roughly speaking, is a measure of the numbers of ``holes'' or handles in the surface. orientability was already discussed intuitively. We closed this section by giving it a bit more rigorous treatment.

Let S be a surface considered as a subspace of Rⁿ for some n. Let $\omega$ be a differentiable closed curve on S. In analytic terms, if the curve is parametrised as a differentiable function

$$\omega$$ : [0, 1]$$\rTo$$S $$\subseteq$$ Rⁿ,

then $\omega$ has components ($\omega_{1}^{}$,...,$\omega_{n}^{}$), with $\omega_{i}^{}$ : [0, 1]$\rTo$R a differentiable function for each i. Then the derivative $\omega{^\prime}$ is again a function from [0, 1] to Rⁿ given by $\omega{^\prime}$ = ($\omega{^\prime}_{1}$,...,$\omega{^\prime}_{n}$) and for each t $\in$ [0, 1], $\omega{^\prime}$(t) is a tangent vector to $\omega$(t).

Consider the tangent plane to the surface and at some point $\omega$(t) on the curve $\omega$. This plane can be obtained as the linear span of the tangent vector $\omega{^\prime}$(t) and the tangent vector to a different curve, say $\alpha$ at the same point. Choose a normal vector n at the point $\omega$(t), i.e. a unit vector perpendicular to the tangent plane at the chosen point. Now consider the system given by $\omega{^\prime}$(t) and n at this point. One can think of moving this system along the curve continuously. That is to say, fix the direction on the curve in which you are moving and don't ``flip'' the normal vector to the ``other side''. If by the time you get back to the starting point you have the same system you started with and this property holds for every smooth closed curve on the surface, then the surface is called orientable. Otherwise it is called non-orientable.

Exercise 3.5   Convince yourself that a closed orientable surface of any genus is indeed orientable, with respect to the above definition of orientability, and that the Projective plane and the Klein Bottle are not orientable.