I'm trying to work out an answer for my previous question and I'm stuck with the following issue:
In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath and Vipismakul, in the example at the bottom of page 8 they are considering the moduli space $$\overline{\mathcal{M}}_{(0,2)(0,\langle 1,1,\rangle)}$$ which studies the configurations of an annulus with two boundary marked points, $1$ for each boundary component. They conclude that "it too is not a CW-complex, again due to a loop of weight two".
On the other hand, Abouzaid in his paper "A geometric criterion for generating the Fukaya category" , in appendix C.4 (page 38) claims that the compactified moduli space $$\overline{\mathcal{C}}^-_1$$ is a manifold with boundary (actually diffeomorphic to the interval $[0,\infty]$). For reference, $\overline{\mathcal{C}}^-_1$ is the compactification of $\mathcal{C}^-_1$ which is the real line consisting of the annuli $\{z\in \Bbb C \mid 1 \leq |z| \leq r\}$ and marked points $\{1, -r\}$.
I'm sure I am missing something but in the first example we are dealing with a $2$-dim space, and in the second one with a segment.
What are the additional condition in Abouzaid's moduli space that justify his model for the compactification? The only apparent difference between the two is that in $\mathcal{C}^-_1$ we are forcing a symmetry in the configurations. Does it mean $\mathcal{C}^-_1$ is the fixed point set of a certain $\Bbb Z_2$ action on $\mathcal{M}_{(0,2,\langle 1,1\rangle)}$ which happens to be a $1$-manifold?
My end-game is to prove that this moduli space of annuli with an additional symmetry is a manifold (with corners), so I'm trying to understand how these spaces are built. This question is closely related with these two old questions: (an old question of mine and this question on the compactification of the moduli space of annuli.)
Thanks for any sort of input/observations!
