# Questions tagged [manifolds-with-boundary]

For questions about manifolds with boundaries, as well as manifolds with corners, and other such generalisations of the notion of a manifold.

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### $M\setminus\partial M$ open and proof of dimension

Let $M$ be an $n$-dimensional manifold with boundary. Show that: $M\setminus\partial M$ is an open subset of $M$ and it is an $n$-dimensional manifold, and that $\partial M$ is a closed subset of $M$...
1answer
257 views

### Do trivial homotopy groups imply existence of boundary preserving homotopies?

Let $N$ be a smooth $d$-dimensional connected orientable manifold $N$ which have the following property: For every smooth $d$-dimensional manifold $M$ with non-empty boundary, and for every smooth ...
1answer
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### Is uniform limit of embeddings injective?

$\newcommand{\pl}{\partial}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be smooth compact, connected, oriented manifolds of the same dimension with non-empty boundaries....
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### Intersection of a minimal surface and a closed ball

Let $u:\Omega\subset\mathbb{R}^2\to\mathbb{R}$ such that $u$ satisfies the differential equation of minimal surfaces on the region $\Omega$. Let $p\in(\Omega\times\{0\})\subset\mathbb{R}^3$ and a ...
0answers
144 views

### definition of essential annulus contained in the boundary

I read John Hempel's paper "3-manifolds viewed from the curve complex". In the proof of Theorem 3.2, he calls annuli in the boundary of a solid torus essential. (see picture below from his article). ...
1answer
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### There is no diffeomorphism between one quadrant in the plane and the half plane

I am trying to prove rigourosly that the unit square $[0,1]\times [0,1]\subset \mathbb R^2$ is not a differentiable manifold with boundary (I've searched for a rigouros proof, but haven't found anyone)...
1answer
130 views

### Is there any necessarily NON-simply-connected bordism between compact 1-manifolds?

As I seem to have confirmed in this previous question, any compact 1-manifold is homeomorphic to the (possibly empty) disjoint union of circles. So my question reduces to essentially: Does there ...
0answers
205 views

### Is the boundary of every 2-manifold the disjoint union of circles?

Note that in the title I am being sloppy and should say "Is the boundary of every 2-manifold with boundary the disjoint union of circles?". Also when I say "disjoint union of circles" I mean up to ...
1answer
179 views

### Long exact sequence for manifolds with boundary

Let $T^* S^n$ be the disk cotangent bundle on $S^n$ consisting of covectors with norm less than or equal to $1$. I am confused about the long exact sequence of the pair $(T^*S^n, \partial T^*S^n)$. ...
0answers
196 views

### Manifolds with boundaries and partitions of unity

How do I 1 show that $M=[0,3]\subset \mathbb{R}$ is a manifold with boundary? 2 find a $C^2$ partition of unity for the open cover $M=[0,2)\cup(1,3]$? 3 show that $\omega=(x-2)dx$ is/is not an ...
1answer
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### Does Stokes hold? (Manifolds)

Let $M$ be the manifold with boundary $M=\mathbb{R}_{\geq0}$ and $\omega\in\Omega^0(M)$ a 0-form. Suppose both $\int_M d\omega$ and $\int_{\partial M}\omega$ are finite. Does Stokes theorem hold? ...
0answers
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### Mobius band gluing construction

Can somebody help me understand how the mobius band can be viewed as $\mathbb{R}\times [0,1]/\sim$ where $(x,y)\sim (x+1,1-y)$? (also, if somebody can help me with the appropriate tags)
2answers
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### Does smoothness on the interior and on the boundary (separately) imply smoothness?

Let $M,N$ be smooth manifolds with boundary. Suppose we have a map $\phi:M \to N$ which satisfies the following properties:  (1) \, \, \phi(\operatorname{int}M)=\operatorname{int}N,\phi(\partial M)...
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### A local diffeomorphism can map a boundary point to an interior point

I would like to find an example for a local diffeomorphism between smooth manifolds with boundary which maps some boundary point to an interior point. I am quite sure such an example exists.
3answers
269 views