We start with this Fox-Artin example:
As we saw in this previous post: this arc contains just one wild point; hence its complement is simply connected. But we can say more than that: this arc is “cellular”: that means that there is a collection of smooth 3-balls where . That isn’t too hard so see: start by taking a small round 3-ball that encloses the wild point and taking a union of that ball with a tubular neighborhood of the rest of the arc outside of the ball. That first gadget is “almost” a ball; it might have a few handles which can be filled in. Then take a smaller ball around the wild point and repeat the process with a smaller tubular neighborhood…and repeat.
In fact, we can do this with ANY arc that has just one wild point.
Therefore which is homeomorphic to ; hence this wild arc complement is homeomorphic to the complement of a tame arc.
Aside: this shows that arcs are NOT determined by their complements whereas smooth (or p. l. ) knots are.
But this arc is still wild; it has penetration index 3 near the wild point.
Now consider this example
This is like the wild Fox-Artin arc talked about in the previous post in this series, except that “one stitch has been missed. It’s complement is simply connected; the proof of this fact is suggested by this diagram:
The small loop represents a generator of the fundamental group; note how it bounds a disk in the complement of the arc.
However the complement is NOT ! There are different ways to see that; here is a fun one:
This shows with a smooth simple closed curve in its complement. Note: is compact (it is a smooth knot). Therefore there are two different ways to use to show that is NOT homeomorphic to :
1. In , every compact set lies inside of some ball . But one can show that there is no ball that contains and misses
2. One can use algebraic topology: the fundamental group of is NOT finitely generated; in , the complement of any smooth knot is finitely generated.
This gives an example of an open, simply connected manifold that is not homemorphic to .
The interested reader can consult the following references for more detail:
R. Daverman and G. Venema: Embeddings in Manifolds, American Mathematics Society Graduate Studies in Mathemtics, Vol. 106, 2009: Section 2.8
T. B. Rushing, Topological Embeddings, Academic Press, (Pure and Applied Mathematics, Volume 52) 1973, Section 2.4.