I think that Mazur's construction need needed the embedding to be C1 at one point! Victor On Fri, Oct 9, 2015 at 3:11 PM, Dan Asimov <asimov@msri.org> wrote:
On Oct 9, 2015, at 11:54 AM, Victor Miller <victorsmiller@gmail.com> wrote:
Yes indeed, this argument is due to Barry Mazur. It's contained in his PhD thesis from Princeton (in 1958 I believe). It's only about 6 or 7 pages long! The story is that there was a seminar being given at Princeton about the attempts to prove the generalized Schoenflies conjecture (that if you embed an n-sphere into an n+1-sphere, that the "inside" and "outside" are both homotopic to a point). After the introductory session Mazur went home to think about it, and came up with the above proof.
As I learned from the topology textbook by Hocking & Young, the "Alexander horned sphere" as in this illustration:
https://www.pinterest.com/pin/330803535106498064/ <https://www.pinterest.com/pin/330803535106498064/>
shows that an embedding of the n-sphere into an (n+1)-sphere better be differentiable for the Schoenflies theorem to have any chance of being true!
—Dan
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