Geometric topology – Wild character of the codimension 1 subvarieties of the Euclidean space

This question arose from this pile trading post. I am writing a thesis on the $ s $-Cobordism and Siebenmann's work on the end obstructions. Combined, they give a quick proof of the uniqueness of the smooth structure for $ mathbb {R} ^ n $, $ n geq $ 6. Siebenmann's theorem says pretty much that for $ n geq $ 6 a contractible $ n $-collecteur $ M $ which is simply connected to infinity integrates (smoothly) inside a compact collector. Since this compact variety is contractible, by the $ s $-cobordism, it is diffeomorphic to the standard $ n $-disk $ D ^ n $ (see Minor Conferences on the $ h $-cobordisme for example). It follows that $ M = text {int} D ^ n $ is diffeomorphic to $ mathbb {R} ^ n $.

The problem is that the case $ n = $ 5 is not covered. I am aware of Stallings beautifully written On the piecewise linear structure of the Euclidean space but I'm looking for a way to deal with the $ n = $ 5 case via Siebenmann's end theorem and the good $ s $theorem of -cobordism (see link to the question mse). This brings me to the next question, which is interesting in itself

Since codimension 1 is well integrated and smooth $ S subset mathbb {R} ^ {n + 1} $, is there a diffeomorphism auto $ mathbb {R} ^ {n + 1} rightarrow mathbb {R} ^ {n + 1} $ who wears $ S $ in a region limited to one dimension $ mathbb {R} ^ n times (-1, 1) $ ?

Now if $ M $ is a multiple that is homeomorphic to $ mathbb {R} ^ $ 5, the product $ M times mathbb {R} $ is homeomorphic to $ mathbb {R} ^ $ 6, and therefore also diffeomorphic. Subject to the existence of diffeomorphism in my question, we could find a diffeomorphism $ f: M times mathbb {R} rightarrow mathbb {R} ^ $ 6 that cards $ M times $ 0 in $ mathbb {R} ^ 5 times (-1, 1) $. This would produce a good $ h $-cobordism between $ M $ and $ mathbb {R} ^ $ 5 taking the area between $ f (M times 0) $ and $ mathbb {R} ^ 5 times $ 1 in $ mathbb {R} ^ 5 times mathbb {R} $. Since $ M $ is simply connected, the good $ s $-cobordism theorem applies and shows that $ M $ and $ mathbb {R} ^ $ 5 are really diffeomorphic.

Aggressive geometry – Singularity of Brill-Noether subvarieties of smooth curve varieties of Picard

assume $ C $ is a smooth projective curve on complex numbers. The singularities of the theta divider $ Theta $ in $ Pic ^ {g-1} (C) $ is described in the literature. It is $ W ^ {1} _ {g-1} = {l in the image ^ {g-1} (C): h ^ 0 (l) geq 2 } $.
I was able to find in the literature the expected results of the dimension.
Are the singularities of $ W ^ 1_ {g-1} $ known.

Question: What is the dimension (exp) of $ Sing (W ^ 1_ {g-1}) $, for a generic curve?

Riemannian geometry – Are complete minimal sub-varieties closed?

It depends on what you mean by submultiple (at least for surfaces in three-dimensional ambient spaces).

Nadirashvilli builds an example of complete minimal immersion in the unit balloon in $ mathbb {R} ^ $ 3. In particular, immersion is not appropriate and the image is not a closed subset of $ mathbb {R} ^ $ 3. On the other hand, Colding-Minicozzi has shown that any integrated minimum surface $ mathbb {R} ^ $ 3 (ie a two-dimensional subvariety of $ mathbb {R} ^ $ 3 in the usual sense) that has finite topology must be properly incorporated and therefore be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of unresolved issues about the general nature of this phenomenon – for example. is this true for finished genre surfaces? Any integrated minimal surface?

dg.differential geometry – The growing union of incorporated subvarieties is a submerged variety

While working on the proof of the stable variety theoremI came across a problem that I am not able to really grasp. Given Anosov $ f: M $ to M $ on a compact Riemann collector $ M $, we can prove that for $ epsilon> $ 0 small enough for each $ x in $ M, $ W ^ s_ epsilon (x) $ defined as the set $ {y in M: d (f ^ n (x), f ^ n (y)) < epsilon } $ is $ C ^ 1 $ integrated subtype of $ M $. If we define the 'stable overall variety' $ W ^ s (x) = {y in M: d (f ^ n (x), f ^ n (y)) rightarrow_n 0 } $, so we have this:
begin {align *}
W ^ s (x) = bigcup_n f ^ {- n} (W ^ s_ epsilon (f ^ n (x))
end {align *}

From this statement, I saw several references (for example, Shub & # 39; s Overall stability of dynamic systems instantly conclude that $ W ^ s (x) $ is an immersed submanifold of $ M $. My questions are:

  • what is it immersed for the dynamists? As far as I'm concerned, $ N subset M $ is an immersed submanifold of $ M $ if $ N $ can be endowed with a smooth structure such as the map of inclusion $ iota: N $ M is a soft immersion, identical to a local integration.

  • Looking at the problem here, we see that $ W ^ s (x) $ is in fact given by a growing union of integrated subvariety. It is true that a growing union of integrated sub-varieties is a submerged variety? If so, do we have references for this claim?

Thank you.