The definition of simplicial complexes allows decomposing the above figure by $2$-simplicies $[a,b,c],[c,d,a]$.
But the definition of $\Delta$-complexes in Hatcher's algebraic topology does not allow such decomposition, since such decomposition forces both $[a,c],[c,a]$ to be contained in the $\Delta$-complex, which violates the requirement that all the points in the complex should be contained in exactly one open simplex.
This happens because when we construct simplicial complexes, we glue and then orient the simplicies, while for $\Delta$-complexes, we orient and then glue, so that distinct $[a,c]$ and $[c,a]$ cannot be glued together.
This does not seem to be a problem in this simple case because we can decompose the figure by $[a,b,c],[a,c,d]$ and this satisfies the definition of $\Delta$-Complexes (Note that $[c,d,a]$ and $[a,c,d]$ are distinct because Hatcher does not identify simplices up to even permutations). But my question is, does this also work for more complicated cases? That is, does every simplicial complex become a $\Delta$-complex by giving suitable orderings for each simplex? (The collection of simplicies should remain the same if the orderings are ignored.)
Best Answer
Instead of ordering the vertices of each simplex separately, just pick a single total ordering on all the vertices of your simplicial complex, and then restrict it to get an ordering of the vertices for each simplex. This will then make every face of each simplex have its vertices in the same order as the original simplex, so that you get a $\Delta$-complex.