## 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.