ScalingStacks

Remark 7.2.5. Let $Z$ be a closed subspace of ${\mathbf{R}}^{N}$ for some $N>0$. Assume there is a finite subset $E$ of $Z$ such that $Z-E$ is a $1$-dimensional submanifold of ${\mathbf{R}}^{N}$ with no boundary and such that given $e\in E$, there is $n^{\prime}_{e}>1$ and a finite family $\{j_{e,i}\}_{1\leq i\leq n^{\prime}_{e}}$ of smooth embeddings $j_{e,i}:(-1,1)\to{\mathbf{R}}^{N}$ such that

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  $j_{e,i}(0)=e$,

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  $j_{e,i}((-1,0)\cup(0,1))\subset Z-\{e\}$,

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  $j_{e,i}((-1,1))\cap j_{e,i^{\prime}}((-1,1))=\{e\}$ for $i\neq i^{\prime}$

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  ${\mathbf{R}}\frac{dj_{e,i}}{dt}(0)\neq{\mathbf{R}}\frac{dj_{e,i^{\prime}}}{dt}(0)$ for $i\neq i^{\prime}$ and

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  $\bigcup_{i}j_{e,i}(-1,1)$ is an open neighborhood of $e$ in $Z$.

Let us choose in addition an open subset $Z_{o}$ of $Z$ containing $E$ and an orientation of the $1$-dimensional manifold $Z_{o}-E$. We assume that $Z-Z_{o}$ has finitely many connected components, none of which are points. We assume furthermore that given $e\in E$ and $i\in\{1,\ldots,n^{\prime}_{e}\}$, the orientation of $j_{e,i}^{-1}(Z_{o}-\{e\})$ extends to an orientation of $j_{e,i}^{-1}(Z_{o})$.

Given $e\in E$, we denote by $\iota$ the involution of $C(e)$ that swaps $j_{e,i}((-1,0))$ and $j_{e,i}((0,1))$ for $1\leq i\leq n^{\prime}_{e}$. Note that $Z_{exc}=E$ and $n_{e}=2n^{\prime}_{e}$ for $e\in E$. This defines a structure of curve on $Z$ that does not depend on the choice of the maps $j_{e,i}$.

We leave it to the reader to check that any curve is isomorphic to a curve obtained by such a construction.