(Working on) a non-algebraic proof that S_n is not contractible

For a blog which tries to add something to philosophy as well, its pauses and unfinished threads strike me as less desirable…sorry! I haven’t forgotten about the threads, but other more pressing matters seem to crop up continually.

In the meantime I think I found a nice non-algebraic proof that S_n is not contractible. The proof only uses some continuous-function theory, specifically the Michael Selection Theorem. I’m trying to expand this to a non-algebraic non-simplicial proof of the general Jordan Curve theorem. The first part I will write down somewhere next month, I hope.

Update: unfortunately I made a mistake in my proof-sketch, which I discovered when (finally) writing the proof down. After trying to fix it, I discovered a counterexample to one of its essential arguments. Since mistakes can be a little illuminating also, let me indicate the trouble:

Trouble:
There is a (simple) continuous function f from [0,1]\times [0,1] to [0,1] such that for all x\in [0,1] we have f(x,0)=0 and f(x,1)=1 and yet there is no continuous function t from [0,1] to [0,1] such that for all x\in [0,1] we have 0<f(x,t(x))<1.

Perhaps a nice challenge to find such a function f yourself… it’s not really hard :-).

Spoiler: (don’t read if you want to find a solution yourself) Picture the square [0,1]\times [0,1], in this square draw a bold S smack in the middle (the letter should have some breadth, say b). Elongate the  lower tip of the S toward a b-breadth interval around the point (0, \frac{1}{2}) and the upper tip toward a b-breadth interval around the point (1, \frac{1}{2}). Now let the continuous function f be such that f assumes the value 1 on all the points in the square above the elongated S, and 0 on all the points in the square below the elongated S, and a suitable gradient in between on the points of the elongated S itself.

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About fwaaldijk

mathematician (foundations & topology in constructive mathematics) and visual artist
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