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The familiar joke is that a coffee mug is topologically equivalent to a donut.

I own a coffee mug that is essentially a regular coffee mug with a tube going the middle. I'm not sure how to describe it, but this youtube video shows someone handling a coffee mug just like it.

My friends and I cannot decide whether or not this is homeomorphic to a 2-fold torus or a 3-fold torus or even some other shape, and I lack the math skills to figure it out myself.

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Not necessarily homeomorphic, but it is homotopic to a triple torus aka a pretzel. – fixedp Jul 30 '14 at 11:11
A $3$-fold one. It has the normal hole, the one created by the "bridge" and the one inside the bridge itself. – Darth Geek Jul 30 '14 at 11:11
You could also settle the argument with some clay! – knedlsepp Jul 30 '14 at 11:21

More precisely, we are talking abiout the surface of the mug. Without this extra, this would be a simple torus. If the tube were just a filled bar, that would make one extra handle. "Drilling" a hole through that bar adds another handle. So what we have is a three-fold torus (three donuts smashed together). It's a fun mental exercise to deform the mug to "separate" the handles.

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Can you make a picture of it for us non mathematicians? This question makes me want to make the donut to complement the mug, but I don't know what a three-fold torus looks like. Is it a normal 3d structure, can it be made from dough? – rumtscho Jul 30 '14 at 15:09

It is a surface of genus 3.

First disregard the outer handle - we can always just add that back on later. We then have to show that the middle bit has genus 2.

To see this, we flatten the cup out, so it looks like a pancake with a thick handle. I'll show how we can go from a pancake surface (a flat sphere) to our surface by adding two handles.

First, we add one handle going over the top, not cutting through the pancake shape. If you made this from clay, it would have a solid interior. Next, you drill out the inside of this solid handle, which is the same as adding a second handle going through the first. This process has added two handles, so counting the original one we removed, we see that the surface has genus 3.

enter image description here

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