I have a pair of Nike Elite socks that have a hole on the heel and another hole at the ball of my foot. Here is the question:
Topologically how many holes are in my sock?
My friend and I have been arguing about this all night.
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2 Answers
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The answer depends on the structure you put on the new sock.
If you think of the sock with no holes as a manifold with the cuff as its boundary it has no holes. Then a new hole in the heel creates one and another in the toe a second, for two holes.
If you think of the sock with no holes as a sphere with a hole, then one new hole makes a tunnel (as @Aaa_Lol_dude says) and another new hole makes three.
If your definition of "number of holes" is the rank of the first homotopy group then the first calculation is correct and the second is not. Both the sphere and the sphere "with one hole" have trivial first homotopy group.
There is a way to think of the sphere as a space with one hole: its second homotopy group has rank 1. There is a two dimensional hole because the interior of the bounding sphere isn't there. Then cutting out a circle to create a sock does change the homotopy - it reduces the rank of the second homotopy group from 1 to 0 while leaving the first rank at 0. The general principle is that creating a new hole in a dimension either either adds one hole in that dimension or subtracts one hole in the next higher dimension.
So this controversy is a good introduction to homology.
See https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
answered Jan 20 at 0:50
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3$\begingroup$ I do not quite get the second reasoning. I thought that a sphere without two points would still only have one hole since $S^{2}\setminus\{pt\}$ is homemorphic to $\mathbb{R^{2}}$. $\endgroup$– user649348Jan 20 at 0:59
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$\begingroup$ @AndréArmatowski Nice observation. See my edit. $\endgroup$ Jan 20 at 1:32
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$\begingroup$ The sock in question seems to be made of a vowen fabric; this suggests that the controversy is also a good introduction to persistent homology. $\endgroup$ Jan 22 at 23:46
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Let us see the case where there are no holes in the sock. Then, we, topologically, have no holes in the sock (including the cuff).
The case where we have one hole, that hole joins the cuff to form a tunnel, which means that there are two holes in the sock. With one extra hole, it joins the tunnel, to give you 3 holes.
