From our two balls model, it is easy to see that we can shrink the equatorial inside , until it is a small sphere surrounding the north pole or the south pole.
Let , and let such that . Then varying in the range gives a continuous family of inside . When or , the sphere is shrunk to a single point. When is close to 1 or -1, it's a little sphere around either the north or south pole. When is near zero, it is nearly equatorial.
If we now consider the two regions into which the is divided by this moving , this might seem very similar to some process of turning the ``inside out.'' However, this is not quite correct. We might look at the inside at some point and label the two regions as ``inside'' and ``outside'' based on which region was smaller. We do this because we are used to Euclidean space, where the ``inside,'' of a closed region usually appears smaller, and the outside usually includes the point at infinity.
However, the region inside the sphere is already finite, and so calling this process ``inverting,'' or swapping the ``inside'' and ``outside'' of inside really doesn't make a whole lot of sense. All we can say is that we can distort inside in such a way as to shrink one region down to a point while expanding another region from a point to the whole sphere.
Following from this, we see that any closed surface inside can be ``inverted,'' by simply gluing to a small region of the manifold, and then ``inverting'' this sphere.
From now on I'll use the word ``inversion'' in the sphere to mean a process like this which apparently reverses the two regions in created by a dividing surface in .
Notice that and are identical manifolds. So we know, trivially, that we can invert inside .
Notice also that we can embed the flat torus, , inside . To see this best, consider our space to be . Then the sphere consists of all points such that , and the torus consists of all points such that .
Inside , this torus can be ``inverted,'' that is, it can be deformed in such a way so that (1) the generators of the torus are swapped, and (2) the apparent ``outside'' is swapped with the apparent ``inside.'' Here are a series of pictures to illustrate the second point; pictures showing the generators being swapped are in section 4.2.
The important point to notice from this is that divides into two regions, both of which are topologically equivalent to a filled . It is by no means obvious that this is a general fact about inside . Furthermore, we can ask whether there is any deeper structure behind this inversion, or whether it is merely a manifestation of the fact, noted above, that any manifold can be inverted inside of the sphere.
We know that any manifold can be inverted inside by gluing it to a sphere. If we know that in general can be embedded inside , then we will know that can be inverted inside , by gluing the torus to a and inverting that .
The proof is inductive. To begin with, we know that can be embedded in , which can be stereographically projected to .
If, in general, can be embedded in , then we can embed in . Think of this simply as as a ``thickened'' or ``fattened'' torus. Next, we cross both with , to get . Now, we can draw a little circle in to see that . And, of course, . So . Since can be embedded in , can be embedded in as we can just project to via stereographic projection.
So, in general, can be embedded and ``inverted'' inside . The next question is whether the neat symmetry of in generalizes as well. To do this we will need to look closer at the structure of .