Hi,I want to show that the set of boundary points on a manifold

[seydunas]

Hi,

I want to show that the set of boundary points on a manifold with boundary is well defined, i.e the image of a point on a manifold with boundary can not be both the interior point and boundary point on upper half space. To do this, it is enough to show that R^n can be homeomorphic to upper half space. How can i show this fact?

Best regards

 

[Office_Shredder]

I would suggest starting by just proving R is homeomorphic to the positive real numbers, and seeing how you can generalize that result

 

[seydunas]

Sorry,

This is not true. It will be R^n can not be homeomorphic to upper half plane. Note that upper half plane is {x=(x_1,,,,,x_n) \in R^n : x_n =>0}.

 

[quasar987]

Try this: if you remove a point on the boundary of the upper half plane, show using Mayer-Vietoris that you do not change its homology, but if you remove a point from eucidean space, this becomes a homotopy sphere and so the homology changes.

 

[Bacle2]

How about invariance of dimension? Interior points have 'hoods homeomorphic to ℝ^n-1 , while interior points have 'hoods homeomorphic to ℝⁿ. That would then produce a homeomorphism between ℝⁿ and ℝ^n-1.

Maybe now homeomorphic can write a post titled 'seydunas'

 

[lavinia]

Try this: if you remove a point on the boundary of the upper half plane, show using Mayer-Vietoris that you do not change its homology, but if you remove a point from eucidean space, this becomes a homotopy sphere and so the homology changes.

I don't see this argument. Can you elaborate?

 

[lavinia]

In the case of a smooth manifold a differentiable ray starting on the boundary can only leave the boundary in one direction. That is, its tangent vector can only point inwards. In an open set it can point in both directions.

 

[quasar987]

Hm, well, if there is a homeomorphism f btw H^n and R^n, then removing a point p on the boundary of H^n and removing f(p) in R^n we get a homeomorphism btw H^n - {p} and R^n - {f(p)}. But this is impossible because H^n - {p} is contractible, but R^n - {f(p)} is not.

 

[quasar987]

How about invariance of dimension? Interior points have 'hoods homeomorphic to ℝ^n-1 , while interior points have 'hoods homeomorphic to ℝⁿ. That would then produce a homeomorphism between ℝⁿ and ℝ^n-1.

Maybe now homeomorphic can write a post titled 'seydunas'

Or seydunasism! :/

The invariance of domain thm say if there is a homeo U-->V with U open in R^n and V in R^m, then V is open and m=n. So yeah, H^n is not open so there is no homeo btw R^n and H^n.

 

[Bacle2]

Quasar987 (ism):

I meant that if a bdry. pt p was part of both the interior of M and the boundary of M,
there would be an open set U containing p that is part of both the interior and the
boundary. This set U would then be homeomorphic to R^n (as part of the manifold), and
homeo to R^(n-1), as part of the boundary. Isn't this correct?

Bacle2(ism)

 

[quasar987]

I don't get how a 'hood of a point on the boundary of H^n is homeomorphic to R^(n-1). A 'hood of a point on the boundary of H² is kind of a half-disk and is thus certainly not homeomorphic to R.

 

[Sina]

You don't need more fancier staff than the knowledge of tangent spaces and diffeomorphisms I think.

Hint: consider the dimension of tangent spaces of boundary of upper halfplane and a interior point of half plane. notice that composition of charts belonging to same differentiable structure when composed gives diffeomorphisms.

correction: if it is not smooth we will have to use homeomorphisms and by the correction quasar has made, tangent space argument is not valid so we don't need tangent spaces.

Indeed consider two charts ψ and ρ (from the manifold to Euclidean space and belonging to the differentiable structure of our manifold M) st for a point p in the boundary of our manifold ψ(p)=b is in the boundary of Hⁿ and ρ(p)=i is an interior point of Hⁿ. Now consider the map taking i to b which is ψ°ρ^-1(i) = p. Since these two charts belong to the same differentiable structure this map is a diffeomorphism so it should be an isomorphism of the tangent spaces. However tangent space of b is n-1 dimensional and that of i is n dimensional. So contradiction.

 

[quasar987]

The tangent space at b is not (n-1)-dimensional.

 

[Sina]

Okay this is where geometric intiution has failed me so I refine my answer. Again take the same construction with ψ°ρ^-1(b) = i (there is a mistake also p should be b). This is a diffeomorphism and around b we can find a small enough nbd U such that ψ°ρ^-1(U) = V is in the interior of Hⁿ which is locally Rⁿ and note tha U is locally Hⁿ. This can not be because U is itself a manifold with boundary and it can not be locally modeled by Rⁿ (using the definition of manifold with boundary, actually I know this is a heuristic arguement, actually I am thinking something for this now). Well okay my last argument has already been made above so I don't need to refine it more. If it is not a smooth manifold replace diffeomorphism with homeomorphism. So I see I have not added anything new to the topic really :p

 

[Bacle2]

I don't get how a 'hood of a point on the boundary of H^n is homeomorphic to R^(n-1). A 'hood of a point on the boundary of H² is kind of a half-disk and is thus certainly not homeomorphic to R.

.

I think the 'hood needs to be a subspace one; what I'm using here is that the boundary of an n-manifold M with boundary BD(M) is an (n-1)-manifold (where I think the topology used in BD(M) is the subspace topology of BD(M) on M ). e.g., if we consider the closed disk D={(x,y) x²+y²≤1} , then the boundary (both manifold- and topological, here) is the set of points BdS:={(x,y): x²+y²=1} . This is a manifold (without bdry, since bdry. of a mfld. with bdry is empty) , but with topology given by open sets from the subspace inherited from D, i.e., open arcs in D.

 

[Bacle2]

Never mind my previous post; I mean, it is true that the boundary of an n-mfld with boundary is an n-manifold, is an (n-1)-manifold with subspace charts, but this won't help proof what Seydunas wants.

 

[Bacle2]

Quasar987 wrote, in part: (Sorry, my quote function is not working)

"Originally Posted by Bacle2 View Post

How about invariance of dimension? Interior points have 'hoods homeomorphic to ℝn-1 , while interior points have 'hoods homeomorphic to ℝn. That would then produce a homeomorphism between ℝn and ℝn-1.

Maybe now homeomorphic can write a post titled 'seydunas'

Or seydunasism! :/

The invariance of domain thm say if there is a homeo U-->V with U open in R^n and V in R^m, then V is open and m=n. So yeah, H^n is not open so there is no homeo btw R^n and H^n. "

That was invariance of dimension; an invariance of domain argument would go like this:
Assume there is a homeomorphism h between R^n and H^n , and then consider the inclusion map i: H^n-->R^n i(x,y)=(x,y) between H^n and R^n . Then the composition
ioh: R^n-->R^n is a continuous injection between R^n and itself, with image H^n, so that
H^n is open in R^n --which it is not.

 

[quasar987]

Right. :)

 

[seydunas]

Ok, this question is in the chapter 7 in Lee's book( this is the submersion, immersion chapter). What is the relation of invariance of domain and chapter 7? Why did the auther ask this question at chapter 7?

 

[quasar987]

Well, that question in Lee is not the same question that you asked! You asked a topological question: "why is H^n not homeormorphic to R^n?" (which is equivalent to asking: "in a topological manifold with boundary why can't a point be both an interior point and a boundary point?" )

The question in Lee is easier, it asks "in a smooth manifold with boundary why can't a point be both an interior point and a boundary point?" And he even gives a copious hint: consider \phi \circ \psi^{-1}:\mathbb{R}^n\rightarrow\mathbb{H}^n followed by the natural inclusion \iota:\mathbb{H}^n\rightarrow\mathbb{R}^n.

This gives a smooth map from R^n into itself. It is easy to see from the definition of chart/boundary chart that the derivative at p of this map has full rank, hence from the inverse function theorem, there is a small 'hood U of p in R^n and a small ball B in R^n around \iota\circ \phi \circ \psi^{-1}(p) on which \iota\circ \phi \circ \psi^{-1} is a diffeomorphism, and in particular, a bijection. But this is absurd because the image of \iota\circ \phi \circ \psi^{-1} never dips below the x^n=0 plane while our little ball B does.