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Cover a table with a tablecloth, crumple it up in the middle (while still leaving the edges hanging over the edge of the table), then stab the folds with a pin. You will almost surely poke an odd number of holes in the tablecloth, because one end of the pin is above the tablecloth, the other is below, and each hole indicates one instance of the pin changing sides (the only exception is if your stab lies perfectly tangent to a fold of the tablecloth).

I suspect that this generalizes. I hope that something like this is true:

Let $X$ be a subset of $\mathbb{R}^n \times \mathbb{R}^m$ that is homeomorphic to $\mathbb{R}^n$. For any $r \in \mathbb{R}^n$, let $D(r) = \{ s \in \mathbb{R}^m | (r, s) \in X \}$. Suppose $D(r)$ is never empty for any $r \in \mathbb{R}^n$. Then $D(r)$ almost always (with respect to Lebesgue measure) has an odd number of elements.

In other words, we have crumpled an $n$-dimensional tablecloth into $m$ additional dimensions (while still leaving it hanging over the edges of our $n$-dimensional table), then stabbed it with an $m$-dimensional pin, hopefully in an odd number of places.

Is this a known result? It smells a lot like Sperner's Lemma (which contains a similar statement about oddness), but I'm not entirely sure how.

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If you fix sufficiently precise hypotheses, this is just the fact that the algebraic intersection number depends only on the homology classes of the submanifolds being intersected. –  John Pardon Jan 22 '13 at 23:24
This is Sard's theorem –  Anton Petrunin Jan 22 '13 at 23:37

1 Answer 1

up vote 15 down vote accepted

Of course there are results implying such things in degree theory and as @unknown (google) mentions.

Our "teaching" proof of Sperner's Lemma here develops this intuition for the purpose of pedagogy. So the connection between what you are talking about and Sperner's lemma is there and simply demonstrated (you'll have to do slight extra thinking in the middle of the homotopy that we construct).

In class every year I present this folding idea through a very nice story that I should have written up but have not:

Can we use maths to design an invisibility cloak for Harry Potter?

Harry Potter has an invisibility cloak made of cloth having very simple properties: It has two sides: Front and Back. We will write F|B and B|F to denote the orientation of the cloth.

  • if an observer looks at the Front side, then she does not see the first object behind the cloth and only sees the second object behind it.
  • if an observer looks at the back side, then she will see the first object behind the cloth (i.e., looking at it from Back side it behaves like glass).

An effective cloak is one in which no one looking at Harry can see Harry but Harry can see everything in front of him.

A. So consider this

                 Severus-Snape       F|B Harry B|F      Draco-Malfoy

Professor Snape sees Draco and Draco sees Snape. Harry is invisible and can see both Draco and Snape.

B. But consider this material with one fold

                 Severus-Snape       F|B Harry B|F B|F      Draco-Malfoy

Draco-Malfoy sees Harry and Severus-Snape sees Draco. Bloody-hell as Ron would say, not only is Harry visible but Snape and Malfoy have figured out that he has an invisibility cloak on.

C. Similarly consider this material with one fold

                 Severus-Snape       F|B Harry B|F F|B      Draco-Malfoy

Draco-Malfoy sees Severus-Snape but Severus-Snape does not see Draco. Snape and Malfoy have figured out that an invisibility cloak is being used. Also Harry cannot see Draco; disaster.

So how can we design an invisibility cloak that makes sure that Harry is not found out even if he runs and that he can also see what's in front of him? Use degree theory, of course, (and thus fixed point theory is important).

Put the cloth around Harry from head to toe as follows

F|B Harry B|F

make sure there are no folds initially. Sew the cloak so that there are no holes in it whatsoever.

If the cloth is folding without ripping (i.e., Harry uses it in the way it is designed) all folds will be of the following form

          Harry B|F F|B B|F     Draco-Malfoy 

Odd numbered and if we let B|F be of orientation +1 and denote by -1 the orientation F|B we see any fold configuration will have +1 as the sum of the orientations of the folds. In particular, Harry is never found out and Harry can see Draco.

A clever student always asks about the seam line of the folds. Well by Sard's theorem you cannot see those.

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Very cool answer! –  Johannes Hahn Jan 23 '13 at 15:26

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