# What is a rigorous proof of the topological equivalence between a donut and a coffee mug?

I've seen this example given numerous times, but have never seen a real proof in a textbook.

• What's your definition of "coffee mug"? ;-) Aug 7, 2011 at 8:43
• ... or "donut" for that matter.
– lhf
Aug 7, 2011 at 11:30
• This isn't really the kind of statement one can attach a rigorous proof to until the meaning of the terms "topological equivalence," "donut," and "coffee mug" are fixed. In my experience the first term can mean homotopy, ambient isotopy, or homeomorphism and it is never really clear which one is meant (sometimes intentionally so). "Donut" could refer either to a torus or a solid torus. And a coffee mug... is it thickened? Do you care about the surface? Etc. Aug 7, 2011 at 14:47
• As far as I'm concerned you don't need a more rigorous proof of this kind of statement (which is clearly meant more as an easily-grasped intuitive example than anything else) than "if you made a coffee mug out of clay, you could reshape it into a donut without breaking or attaching anything." Aug 7, 2011 at 14:49
• why "donut" and not a "bagel"? May 22, 2014 at 16:47

The torus and coffee mug are homeomorphic by the identity map $$\operatorname{Id}_{\mathbb{T}^2}:\mathbb{T}^2\to \mathbb{T}^2$$ ;-).

They are both homeomorphic to a sphere with one handle attached. This is quite clear for the coffee mug (where the handle is precisely the handle.....) and it is easily obtained for the donut (a.k.a. torus) by playing a bit with its representation as a square with opposite edges identified.

• Actually, I wouldn't want to eat one of your two-dimensional donuts :) My coffee mugs are rather three-dimensional as well...
– t.b.
Aug 7, 2011 at 10:59
• Not even if it's filled with cream? :) Aug 7, 2011 at 11:04
• If it has a cream filling, it certainly cannot be two-dimensional anymore... :D Aug 7, 2011 at 11:10
• The point is that the topological equivalence will not preserve any cream filling, as that will flow all over the place unless thoroughly beaten. Aug 7, 2011 at 11:33
• The problem with the question is that neither the coffee cup nor the doughnut has been made that precise. In this case it is very difficult to give and explicit homeomorphism between them. On the other hand if you start with a mug without handle, specified as ever you like then it should be possible to produce a flat disc (thickness that of the mug) and then add a handle. (Is the torus a solid torus or just the surface... ditto for the mug?) Aug 7, 2011 at 11:39

I am assuming that the questioner knows that the question is about either the surfaces or the 3-manifolds in question. A rigorous, yet diagrammatic proof is in our book Knotted Surfaces and Their Diagrams.

• It costs 130\$, any chance we could get a better answer then please read a very expensive book Mar 30 at 19:19