$\def\mc#1{\mathcal#1}\def\seq#1{\langle#1\rangle}\def\bbR{\mathbb R}\def\gt{>}\def\dom{{\rm dom\ }}$In some instances, I have seen an appeal to the concept of "the double" of a smooth manifold with non-empty boundary. [Wikipedia](http://en.wikipedia.org/wiki/Double_%28manifold%29) gives a pure nonsense for this: "Precisely, the double is $M \times \{0,1\} / \sim $ where $ (x,0) \sim (x,1) $ for all $ x \in \partial M $ ." The essential problem here is how to construct pairwise smoothly compatible charts covering the boundary. **I ask whether anyone reading this knows how to do this.**


My natural idea of constructing "the" double would be the following. Let $\mc A$ be an atlas for the given $n-$dimensional smooth compact manifold with boundary. Let $M\_0$ be the "interior" and $M\_1$ the "boundary". Let $\mc A\_0$ contain the charts $\phi:\dom\phi\to\bbR^n$ belonging to $\mc A$ with $\dom\phi\cap M\_1=\emptyset$ , and let $\mc A\_3$ contain the rest. So the functions belonging to $\mc A\_3$ are bijections from some subset of $M\_0\cup M\_1$ onto some set of points $x=\seq{x\_0,x\_1,\ldots x\_{n-1}}\in\bbR^n$ where $x\_0\ge 0$ , and mapping points $m\in M\_1$ to $x$ with $x\_0=0$ .


Then one would take as a generating atlas of **a** "doubled manifold" the set $\bar{\mc A}=\mc A\_0\cup\mc A\_1\cup\mc A\_2$ where $\mc A\_2$ contains the functions $\tilde\phi:(m,{\rm w})\mapsto\phi(m)$ where $\phi\in\mc A\_0$ and the fixed ${\rm w}$ is chosen so that $(m,{\rm w})\not\in M\_0\cup M\_1$ for $m\in M\_0$ . As the elements of $\mc A\_1$ one would take the functions $\bar\phi$ for $\phi\in\mc A\_3$ constructed as follows. Let $P:\bbR^n\to\bbR^n$ be the bijection $\seq{x\_0,x\_1,\ldots x\_{n-1}}\mapsto\seq{-x\_0,x\_1,\ldots x\_{n-1}}$ . Then define $\bar\phi$ so that $\dom\phi\owns m\mapsto\phi(m)$ and $(M\_0 \cap \dom \phi ) \times \{ {\rm w} \} \owns(m,{\rm w})\mapsto P(\phi(m))$ .


Then $\bar{\mc A}$ defines a "continuous atlas for the double", i.e. the chart changes $\psi\circ\phi^{-1}$ are homeomorphisms between some open subsets of $\bbR^n$ for $\phi,\psi\in\bar{\mc A}$ , but they need not be differentiable at points $\seq{0,x\_1,\ldots x\_{n-1}}$ .



The double of a smooth manifold with boundary?
I believe that the usual remedy is a collar. That is, for any smooth manifold there is a suitable diffeomorphism from a neighborhood of $\partial M$ to $[0,1)\times \partial M$, or in other words a smooth embedding $[0,1)\times \partial M\to M$ that is "the identity" on the boundary. This allows you to glue along the boundary and get a smooth manifold. To see that the result is independent of the choice of an embedding you use the fact that any two such embeddings are smoothly isotopic.


