# 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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### If the symbol $\partial \Omega$ is used to represent the boundary of $\Omega \subset \mathbb{R}^2$, is smooth or differentiable a pre-requisite?

If we can use the symbol $\partial \Omega$ to represent the boundary of $\Omega$, for instance $\Omega$ is in $\mathbb{R}^2$ ($\mathbb{R}^3$), and thus $\partial \Omega$ is a curve (surface), do we ...
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### Finding an orientable surface with a given boundary

I'm going over the problems here, and I'm currently stuck on 6G-3. Intuitively, I try extend the curve $C$ to a short cylinder and try to pull it out where the two loops meet. However, I find this ...
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### Is the disc minus two holes homotopy equivalent to the disc minus a wedge sum of two holes?

There is an obvious homotopy (given by the usual inverted pant diagram) between the identity map from disc to disc and the map that deforms the separate holes to their wedge sum. But this does not in ...
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### Prove every compact $1$-manifold with boundary always has an even number of boundary points.

There's a claim that Milnor makes in his book Topology from the Differentiable Viewpoint Every compact $1$-manifold with boundary always has an even number of boundary points. I'm not quite sure ...
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### If manifolds $M_1$ and $M_2$ are connected, $M_1\# M_2$ is connected.

Show that if $M_1$ and $M_2$ are connected $n$-manifolds and $n>1$, then $M_1 \# M_2$ is connected. $M_1 \# M_2$ is the connected sum of the two manifolds. This is problem 4.18(b) from Lee's ...
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### $T_P(M)$ vs. $T_P(\partial M)$

I tried to understand in what way an orientation on $M$ induces an orientation on $\partial M$. Note that I am learning in the context of the extrinsic definition of manifolds, i.e. everything is ...
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### Manifolds with corners

Here is what I'm experiencing. A text introduces manifolds. Then, manifolds with boundary is introduced. After all, manifolds with corners is introduced. First of all, I checked whether Implicit ...
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### When is a continuous map $f: M \longrightarrow N$ between smooth manifolds homotopic to a smooth one?

I know there have been similar questions here, but I haven't been able to completely pin down the precise conditions on $M$ and $N$. I have seen one proof of this that uses tubular neighborhood ...
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### $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$

Consider $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$ Where $f^{[-1}]$ denotes the functional inverse of $f$. How to find $f$ ? How about the more General idea of finding $f$ for a given $g$? ...
If you have a Riemannian manifold $(M,g)$ (maybe with other assumptions as need), is there a natural way to extend it to a smooth manifold with boundary? For example, the Lobachevsky space viewed as ...
The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...