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Topology Illustrated -- Addenda

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Let's review the following exercise. A $3\times 3 \times 3$ cube is made from $27$ smaller cubes and then $7$ cubes are removed: one from the middle of each face and the one in the middle of the cube. How many holes does it have now?

Cube with holes.png

Below is a paraphrase of how we were trying to answer the question and, at the same time, figuring out what "hole" means.

-- Is it 1 hole?

-- Hmm, no... Don't think "rooms", think "tunnels".

-- 3? We can just drill 3 tunnels...

-- Hmm, no... Don't think how many tunnels you drill but how many "windows" you create.

-- 6? We have 6 openings to the outside...

-- Hmm, not 6... but close, it's 5. You see, one of the windows doesn't count!

-- Why not?!

-- Hmm... Try to flatten the whole thing.

-- What for?

-- Well, you should have the same number of holes after you flatten it.

-- The bottom "window" stretches out and become the outer border...

-- Right.

Cube with holes flattened.png

Exercise. Once we know how many holes we have, can you point them out?

In case this seems arbitrary, keep in mind that there is another way to arrive at this result. We can take the data about all the vertices, edges, faces, and how they are attached to each other and feed it into a machine. This machine, called “homology”, will produce the same answer, $5$, every time even if you start with a $5\times 5\times 5$ cube...

Exercise. Count the holes below. Then think about the pattern.

Cube subdivided.png

Another look at counting holes: how many holes does a pair of pants have? A pair. No, not three. One can see this by creating a pair of “topological pants”: take a piece of fabric, puncture two(!) holes for the legs, you then realize that there is no need for another hole because the outer edge goes around your waist.

Topological pants.png