I wrote in February:
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.
And, of course, as such things go, I haven’t written down anything yet. Worse, the expansion to a non-algebraic proof of the generalized Jordan Curve Theorem, has hit a serious snag (with interesting ramifications though).
The snag: I cannot prove even the most basic facts about self-intersecting polyhedra. And I cannot find what I want in the literature either. It seems to my uninitiated eye that many basic questions about polyhedra are still open.
Interestingly, much attention to these questions seems to come from computer science, and especially the imaging sector (computer graphics and visualization).
As a mathematician and visual artist, I like such connections very much!