TITLE: Why is this not a well-defined $\Delta$-complex of the torus?
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: For a $\Delta$-complex the orientation of each simplex is given by a linear order on its vertices, the faces then have to be oriented accordingly. Hence, the $1$-faces of a simplex can never be oriented in a cyclic manner.