* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:
 
* <math>U_\epsilon(\hat\mu)</math> is the '''<math>\epsilon</math>-neighborhood of <math>\hat\mu</math>''', i.e. the set of <math>(\hat T,\underline\ell,\underline x,\underline O,\underline\beta,\underline z)</math> satisfying these conditions:
 
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.
 
** For every Morse-type edge <math>\hat e</math>, <math>\ell_{\hat e}</math> lies in <math>(\hat\ell_{\hat e}-\epsilon,\hat\ell_{\hat e}+\epsilon)</math>. Moreover, for every Morse-type nodal edge resp. edge adjacent to a critical vertex, we fix <math>\ell_{\hat e} = 0</math> resp. <math>\ell_{\hat e}=1</math>.
** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v}</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).
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** For every main vertex <math>\hat v \in \hat V^m</math>, the marked points <math>(\underline x_{\hat v},\underline O_{\hat v})</math> must be within <math>\epsilon</math> of <math>(\underline{\hat x}_{\hat v},\underline{\hat O}_{\hat v})</math> (under <math>\ell^\infty</math>-norm for the boundary marked points and Hausdorff distance for the interior marked points).
 
** Similar restrictions apply to <math>\underline\beta</math>.
 
** Similar restrictions apply to <math>\underline\beta</math>.
 
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.
 
** <math>\underline z</math> must be within <math>\epsilon</math> of <math>\hat{\underline z}</math> in Hausdorff distance.
 
We can now construct the gluing map <math>[\#]</math>.
 
We can now construct the gluing map <math>[\#]</math>.
 
We will not include all the details.
 
We will not include all the details.