Eventually I want to talk about proper knot theory: this is the theory of proper embeddings of the real line into open three manifolds.
Note: a proper embedding is one in which the infinities of the real line go to ends of the manifold; the inverse image of the embedded real line intersect a compact set in the manifold has to have a compact pre image. Example of a non-proper embedding of the real line:
Let ; if is any compact set containing in its interior then is unbounded and therefore not compact.
So it would be helpful to have some nice open manifolds to work with and one of the nicest examples of such gadgets are the Whitehead Manifolds.
This diagram represents three stages of the construction.
Here is the first stage: (imagine thickening the knot inside the torus)
If one keeps iterating these embeddings to get nested solid tori where is situated in as shown, and then one gets which is called the Whitehead continuum. Then is the required manifold.
One can also view this as an increasing union as well: in the “simple” diagram think of the inside torus as stage 1 and the outer torus as stage 2; that is, the inner torus is and the outer is . Now there is a homeomorophism where . Then define (composed times). Then the Whitehead manifold is .
It is well known that is open, contractable, not homeomorphic to and that is homeomorphic to ; there is also a new result by Gabai that says that is the union of two copies of . The eventual goal of this note is to show that is homeomorphic to . To do that, I’ll do a bit of warm up and then work through this paper by Glimm, which can be freely accessed here.
Warm up: why does a smooth knot unknot in 4-space?
When we work through the proof that is homeomorphic to we will be using the “4’th dimension” to lift the inner torus so as to unwrap it and then put it back in a trivial fashion (the inner torus would lie in a ball of the outer torus) and a product of these things is easy to visualize. So we need some 4-dimensional intuition to get the pictures clear in our mind.
(to you experts: yes, I know that can be thought of a cellular in and therefore is homeomorphic to , but I am trying to get an intuitive visualization here)
So, let’s see how a knot unknots in . Sure, there is an easy way: let where is identified with the hyperplane in . Now pick a point and let be the cone of from . is a piecewise linear disk but is NOT locally flat (and therefore isn’t a famous “slice disk”) because the link of the cone point is a copy of the knot , but this is enough to show that unknots. But that is not very satisfying; let’s construct a nice locally flat disk instead.
Start with a knot diagram; the knot consists of a finite number of arcs in the plane with a finite number of “overpasses” which leaves the plane but stays in the hyperplane.
Here is a view from the top with the z-axis coming at you and the w-axis just somewhere else. 🙂
Now any p. l. (or smooth) knot can be changed into the unknot by changing a finite number of crossings (the above example: we can get the unknot by changing any of the crossings). The minimum number that have to be changed is called the “unknotting number”; in general this is hard to calculate. But that doesn’t matter for this exercise. That that means for us is that there is a homotopy for which ,
is the unknot and if is the collection of arcs of on the plane, for all . That is, this homotopy keeps the plane arcs fixed and moves the necessary overpasses into underpasses so as to effect the necessary crossing changes. Note: of course, the knot has to pass through itself in ; we do get double points at a certain stage. In fact, we might say this homotopy affects the coordinate; think of a homotopy that drives the “unknotting overpasses” straight down and fixes the rest of the knot. So let’s describe the homotopy (in ) in two parts: and then . Now the ambient isotopy in can be described by for points on (the set of arcs in the plane) and
for and .
So the upshot: this isotopy swings the offending overpasses into the next dimension and then returns them underneath the plane; in the final slice, the knot has been taken to the unknot.
So, what does this have to do with the Whitehead manifold? Look at the simple picture of the first stage: one does exactly the same isotopy described in to change the single crossing into an under crossing; this puts in in a trivial manner; hence we have a homeomorphism from to where the latter is a trivial torus pair (one torus embedded in the larger torus in a geometrically trivial way (the inner torus lies in a ball contained in the larger torus).
So we obtain a homeomorphism between the pair which is the identity on the boundary and the latter is just a pair of solid tori where lies in a ball in . The reference linked to above (Glimm) shows that such homeomorphisms can be pieced together to yield a homeomorphism and the latter is known to be homeomorphic to .
Basically, the factor gives one enough “room” to unlink the Whitehead type link in the torus.
This best is simply connected; in fact it is contractable. But it has a very weird property: if one, say, constructs a smooth simple closed curve by, say, running along the center line of one of the defining tori, the complement of this curve (the knot group, if you well) is NOT finitely generated! So this beast is NOT .