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:
There is a (simple) continuous function from to such that for all we have and and yet there is no continuous function from to such that for all we have .
Perhaps a nice challenge to find such a function yourself… it’s not really hard :-).
Spoiler: (don’t read if you want to find a solution yourself) Picture the square , in this square draw a bold S smack in the middle (the letter should have some breadth, say ). Elongate the lower tip of the S toward a -breadth interval around the point and the upper tip toward a -breadth interval around the point . Now let the continuous function be such that assumes the value on all the points in the square above the elongated S, and 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.