Writing down homeomorphisms in a formula

Question

Hey I am dealing with this question since some months. How rigorously do I have to prove that two "obviously homeomorphic" spaces are really homeomorphic? Example: Cup homeomorphic to a donut... Is it even possible to write down a homeomorphism in this example?

Answer 1

ZPlaya: The comments suggest that this homeomorphism comes by magic. But the way the proof that `an orientable surface with $\chi(\Sigma) = 2-2g$ is diffeomorphic to the standard $\Sigma_g$' is by constructing an explicit homeomorphism.

There are many ways this is done. Almost all of them start by `triangulating the surface' which is a difficult process but still done explicitly if you chase back all relevant results. This decomposes the surface into triangles. In your case, imagine that your coffee mug, instead of being smooth, was made of hammered gold with triangle-like faces. This gives a combinatorial sense of what's going on with your coffee mug.

We then identify the building-blocks of the surface. In your case, the handle is a copy of $S^1 \times [0,1]$ (the cylinder, or the annulus), and you need to show that the "coffee-mug without a handle" is too. This is harder, but the coffee mug has a portion starting at the lip and going inside homeomorphic to $D^2$, and a portion starting at the lip and going to the bottom, homeomorphic to $D^2$ minus two smaller discs (think: face, where I have removed where the eyes would go; or think pair-of-pants).

Then I observe that pasting these together --- the disc-bits, I mean --- I get what looks like the 2-sphere with two discs removed. This is a theorem you prove when studying quotient topologies, that 'two discs pasted together along their boundary is homeomorphic to a sphere'. Then you are almost done --- you now need to prove that the sphere with two discs removed, with $S^1 \times [0,1]$ (a `handle') glued on to those pieces, is homeomorphic to a torus.

I am being inexplicit here if only because making all of this explicit --- with formulas --- would require massive amounts of piecewise data on each piece I've described above. It is possible, but topologists are content to know the existence of a formula for a continuous function rather than write it down, because the formula rarely provides insight into what is going on. But we do prove that a homeomorphism exists, by proving that every surface can be chopped up into building-blocks, and then proving that the building-blocks can only be pasted together in relatively simple ways.

A very careful proof of this will require every single piece of knowledge you learn in a topology course, so it is not suitable for an answer here. :)

Answer 2

The comments above summarize nicely, but I thought it should be put in an answer.

To describe the homeomorphisms, you first need to describe a "cup" and a "donut" as topological spaces. As Tesla says, a parametrization of the cup will work fine (although it will be very complicated). The homeomorphism would then probably be a composition of homeomorphisms that could be described in terms of the coordinates: one that "lifts up" the base of the mug, one that "squashes" the now solid mug into its handle, etc. The formula you want will probably look horrible, but it's possible to write one down (because, as you said, it exists!).

Gerry's method also works, provided that the donut is hollow and so is the mug: if both are solid (i.e. the donut is a copy of $D^2\times S^1$, which is intuitively "more like a donut than the torus"), then some more work needs to be done, as they are now handlebodies, not surfaces.