Confusion about the delta complex structure of Torus I ran into a confusion while trying to understand delta complex structures on topological spaces. For concreteness, we consider the Torus ($T^2$). I have constructed two delta complex structures on $T^2$ which are only slightly different from each other. These two cases produce the same Cohomology groups, but one of them (the second case) gives the wrong cup product. Here are the two cases.
First Case:
We first define the following delta complex structure (please correct me if I have done something conceptually wrong here).
2-simplices:
\begin{align}
&\sigma_2^U[0,1,2]=U \subset T^2 \\
&\sigma_2^L[0,1,2]=L \subset T^2
\end{align}
1-simplices
\begin{align}
&\sigma^a_1[0,1]=a \subset T^2 \\
&\sigma^b_1[0,1]=b \subset T^2 \\
&\sigma^c_1[0,1]=c \subset T^2 \\
\end{align}
0-simplex
\begin{align}
\sigma[0]=v
\end{align}
Restrictions:
\begin{align}
&\sigma_2^U|_{[0,1]}=b \\
&\sigma_2^U|_{[1,2]}=a \\
&\sigma_2^U|_{[2,0]}=c \\
&\sigma_2^L|_{[0,1]}=a \\
&\sigma_2^L|_{[1,2]}=b \\
&\sigma_2^L|_{[2,0]}=c. \\
\end{align}

Second Case:
In the second case we consider a slightly different version:
2-simplices (same as before):
\begin{align}
&\sigma_2^U[0,1,2]=U \subset T^2 \\
&\sigma_2^L[0,1,2]=L \subset T^2
\end{align}
1-simplices (same as before)
\begin{align}
&\sigma^a_1[0,1]=a \subset T^2 \\
&\sigma^b_1[0,1]=b \subset T^2 \\
&\sigma^c_1[0,1]=c \subset T^2 \\
\end{align}
0-simplex
\begin{align}
\sigma[0]=v
\end{align}
Restrictions:
Here we change the restriction maps as
\begin{align}
&\sigma_2^U|_{[0,1]}=b \\
&\sigma_2^U|_{[1,2]}=a \\
&\sigma_2^U|_{[2,0]}=c \\
&\sigma_2^L|_{[0,1]}=c \textrm { (changed)} \\
&\sigma_2^L|_{[1,2]}=a \textrm{ (changed)}\\
&\sigma_2^L|_{[2,0]}=b \textrm{ (changed)}\\
\end{align}

Both of the structures gives the same cohomology groups. But the second case does not give the correct cup product because the "front face" and the "back face" of $\sigma^L_2$ change in the second case. This means that I have done something illegal in the second case. If someone can point out what is wrong with my second construction, or any kind of other conceptual issues, it will be helpful.
 A: I will assume you're using Hatcher's definition for a $\Delta$-complex. He says, "The vertices of a face, or of any subsimplex spanned by a subset of the vertices, will always be ordered according to their order in the larger simplex," and the problem is that you have an inconsistent ordering on the vertices of the edge $c$. It might be clearer to think of your pictures of attempted $\Delta$-complex structures on a square, and then do the identifications later. From this point of view, in the second picture the edge $c$ is going from vertex $2$ to vertex $0$ in $U$, but it's going from vertex $0$ to vertex $1$ in $L$: the induced order of the vertices from $U$ is not consistent with that from $L$.
A bit more detail: you should view vertex $i$ as the vertex opposite the $i$th face, and that's where my indices are coming from. I find it more helpful to start with an ordering on the vertices and then define the faces: there is a map $d_i$, the $i$th face map, from the set of $n$-simplices to the set of $(n-1)$-simplices, obtained by omitting vertex $i$. These face maps have to satisfy an identity (and this comes from Hatcher's induced ordering):
$$
   d_i d_j = d_{j-1} d_i \text{ for all } i<j.
$$
Your second picture doesn't satisfy this in the case $i=0$ and $j=1$: $d_0 d_1 L = d_0 b = v_{SE}$, the southeast vertex, while $d_0 d_0 L = d_0 a = v_{SW}$, the southwest vertex.
