>>11051427>>11051438I improved the definition and texed it.
>counterexampleI don't fully understand what you mean, it seems like you'd say a full disk and a quarter disk are diffeomorphic.
Details are only supposed to be invariant up to isomorphism, and I Id say there is no linear function that takes a quarter disk (detail of quarter disk in 0) to a full disk (detail of full disk in 0), so that would disprove my theorem.
A further comment: I have to precise more that the way I used diffeomorphic is not entirely correct, I only proved the theorem for whole-R^d diffeomorphisms that take one set to a different one. If the diffeomorphism ends at the border of the set, I cant do my proof.
>flat torusIf something is wavy then the detail will not "converge".
Imagine sth like cos(1/x).
At 0 if you zoom in you will see oscillating waves. All the points in your B1(0) "view" that are touched will just not be in the detail. So the detail here would actually be a point I think.
As you can see details dont work as nicely as one hopes. But I just happened to find a (possible) proof for a useful theorem where *this* definition of detail is good enough.
Stronger definitions of detail could be made.
For example you might require that every point in B1(0) is in the detail of omega or the detail of R^d\omega. So there are no points which keep oscillating between inside and outside as you zoom in.
The ultimate killer would be to use the details of a set and their "configuration" to create a property that, if equal, implies diffeomorphic.
Then diffeimirohism classes = detail configs
But this would probably require a stronger definition of detail and many regularities in the sets and functions.
Would still be awesome though.
