I remember one of my first classes in algebraic topology. The professor was talking about how to prove that . For those who might be rusty: I am talking about the fundamental group of the circle, which is a group structure put on the set of continuous maps of the circle into the circle, where the maps all contain a set “base point” and two maps are equivalent if there is a homotopy (continuous deformation) between the two.
He remarked that he hoped “it was clear” that the circle was NOT simply connected.
That confused the heck out of me, because I had fallen into the trap of confusing the circle with a disk bounded by the circle!
Remember, for years, I had heard things like “the area of a circle is”…when in fact, the circle has area zero. The disk in the plane bounded by the circle has an area though.
So, when I teach, I try to point out bad or inconsistent notation. Example: means rather than as the notation might suggest. But means and NOT . But… means .
And please, don’t even get me started about that appears in integrals. I remember a student asking me about that when we did “integration by substitution”: “we never used the for anything up until now!” he said…correctly.
What got me thinking about this
This blog describes many of the things that I am thinking about at the moment. Currently, I am thinking about “wild knots”, which are embeddings of the circle into 3 space which cannot be deformed (by a deformation of space) into a smooth embedding of the circle.
Here are two examples of knots that can’t be deformed into a smooth knot:
Now the term “knot” implies that an embedding is present; the space that is being embedded is a circle. Of course, one might confuse a particular embedding with the equivalence class of equivalent embeddings; some old time authors distinguished the two concepts. Most modern ones (myself included) don’t.
Now I am interested in knots that are formed by the embedding of two “arcs”, each of which is non-wildly embedded (not wild is called “tame”).
In the case of arcs, authors sometimes mean “the arc itself” and in other cases mean “the embedding of an arc” (e. g. “arcs in 3 space”). Yes, there are some arcs that are so pathologically embedded that there is no deformation of space that takes the arc to a smooth arc. Unfortunately, the term “arc” can mean “the underlying space” or “the embedding”.
This will be one focus of my research in 2015: I hope to show that a knot that has one wild point (roughly speaking: one point that can never be assigned a tangent vector) that is the union of two tamely embedded arcs is never determined by its compliment. That might sound like gibberish, but in 2014 I proved that a knot that is an infinite product of knots (which are converging to a single wild point) has a complement which is homeomorphic to the complement of a knot that is wild at ALL of its points.
Of course THINKING that I can prove something and proving it are two different things. I remember spending two years trying to prove something that was false (I published the counter example) and, for part of my Ph.D. thesis, I attempted to prove something that turned out to be false; of course the counterexample came over 20 years after my attempt.