My lectures notes say that the second diagram isn't a well-defined $\Delta$-complex of the torus because the $2$-simplices aren't totally ordered.
I don't really understand what that means.
Let's call the upper $2$-simplex $U$, the lower one $L$, the vertical $1$-simplex $a$, the horizontal one $b$ and the diagonal one $c$.
If we orient the top $2$-simplex clockwise and the bottom one counter-clockwise. Then the boundary of the top one is $a+b+c$ and the boundary of the second one is $-a-b-c$.
So where's the problem? Why is the first diagram well-defined, and the second one not?