Let $X$ be the quotient space of the disk $D=\{(x,y) \in \mathbb{R}^2 \; | \; x^2+y^2\leq 1\}$ obtained by identifying point on the boundary that are $120$ degrees apart. I'm trying to find a $\Delta$-complex structure on this space. I have the following one, but I'm not sure how to really justify that it is indeed a $\Delta$-complex structure on the space $X$.
My idea for even finding this structure was to first say that the closed unit disk is homeomorphic to the hexagon, and then say that associating the points on the boundary as described above yields the identifications shown in the figure. Is this enough, or do I need to be more technical? Is my $\Delta$-complex even correct?
Thanks!
I would add the symbols for three vertices, to illustrate that the face maps of colored edges of the same color are consistent (they are). Apart of that, for 2D Delta-complexes one should always check that for each triangle the arrows do not "turn round the clock" (in which case the order of vertices is not well defined). As for the further justification, an explicit formula for the six arcs on the boundary of the unit disk corresponding to the external edges of your hexagon could be appropriate.