Articles of manifolds with boundary

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since the 3- (and 4-) handles that close the 0-1-2-handlebody are unique. Is there a good notation for Kirby diagrams for manifolds with […]

Smoothness of the boundary is the only obstruction for being a submanifold with boundary?

Let $M$ be a smooth manifold, and let $S$ be an open smooth submanifold of $N$. Assume the topological boundary of $S$, $\partial S :=\bar S \setminus S$ is a smooth submanifold of codimension 1 in $N$. Is it true that $\bar S$ is a submanifold with boundary of $N$? Somehow, I am not even […]

Manifold Boundary versus Topological Boundary.

Let $M$ a $n$-manifold whit boundary, i.e., for each $x\in M$, there exist $U_x\subseteq M$ open in the topology of $M$ such that $U_x$ is homeomorphic to $\mathbb{R}^n$ or homeomorphic to $\mathbb{H}^n$, where $$ \mathbb{H}^n = \{ (x_1,\ldots,x_n) \in \mathbb{R}^n \;:\; x_n \geq 0\}. $$ Denote by $\partial M $ the boundary of $M$, i.e., […]

Isometric immersions between manifolds with boundary are locally distance preserving?

Let $M$ be a compact, connected, oriented smooth Riemannian manifold with non-empty boudary. Let $f:M \to M$ be a smooth orientation preserving isometric immersion. Is it true that $f$ is locally distance preserving? (in the sense that around every $p \in M$ there exist a neighbourhood $U$ s.t $d(f(x),f(y))=d(x,y)$ for all $x,y \in U$) This […]

(Whitney) Extension Lemma for smooth maps

I am currently reading Lee’s book “Introduction to Smooth Manifolds (2nd edition)”. Corollary 6.27 in that book states that a smooth map $f : A \rightarrow M$ where $M$ is a smooth manifold without(!) boundary and $A \subset N$ is closed (where $N$ is a manifold with possibly nonempty boundary) can be extended to a […]

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward Witten said that Any 3 dimensional manifold can be realized as the boundary of a 4 dimensional manifold. ps. I have noticed […]

Is the square pyramid a manifold with corners?

An n-manifold with corners is topologically an n-manifold with boundary, but with a smooth structure that makes it locally diffeomorphic to $[0,\infty)^n$ instead of $[0,\infty) \times \mathbb{R}^{n-1}$. See also: J. Lee, Introduction to Smooth Manifolds (Chapter 16, Integration on Manifolds) D. Joyce, On manifolds with corners ( The filled cube is naturally a 3-manifold […]

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 orientation on $M$? What I know: Let $M$ be a manifold in vector space $V$. Then it is covered by coordinate patches $f:A\rightarrow […]

The boundary of an $n$-manifold is an $n-1$-manifold

The following problem is from the book “Introduction to topological manifolds“. Suppose $M$ is an $n$-dimensional manifold with boundary. Show that the boundary of $M$ is an $(n-1)$-dimensional manifold (without boundary) when endowed with the subspace topology. So far I’ve manged to prove that the boundary is a second countable Hausdorff space (when endowed with […]