**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.

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