The space $\Delta^n$ with all faces of the same dimension. If the space $A$ is obtained from $\Delta^n$ by identifying all faces of the same dimension; 
What is a $\Delta$-complex structure on the space $A$?
And how can you compute the Simplicial Homology groups on the space $A$?
 A: I think I can compute homology groups for $n=2$. We can understand $\Delta^2$ as chain complex
$$
0 \rightarrow \mathbb{Z}\{f\} \rightarrow \mathbb{Z}\{e_1,e_2,e_3\} \rightarrow \mathbb{Z}\{v_1,v_2,v_3\} \rightarrow 0
$$
$f$ is triangle, $e_i$ are edges and $v_i$ are vertices. 
$$\partial f = e_1+e_2+e_3$$
$$\partial e_i = v_{i+1} - v_i$$
Now we have more choices how to identify elements. For example $e_1=e_2=e_3$ or $e_1=-e_2=e_3$ etc.
Let's do identification $e:=e_1=e_2=e_3$ and $v:=v_1=v_2=v_3$. 
Then $\partial f = e_1+e_2+e_3 = e+e+e = 3e$, $\partial e_i = v_{i+1} - v_i = v-v = 0$. With this identification you get chain complex
$$
0 \overset{0}{\rightarrow} \mathbb{Z}\{f\} \overset{3}{\rightarrow} \mathbb{Z}\{e\} \overset{0}{\rightarrow} \mathbb{Z}\{v\} \overset{0}{\rightarrow} 0
$$
From this we get homology groups
$$
0 \rightarrow 0 \rightarrow \mathbb{Z}_3 \rightarrow \mathbb{Z} \rightarrow 0
$$

One has to be careful with previous identifications. For example for $\Delta^1$
$$
0 \rightarrow \mathbb{Z}\{e\} \rightarrow \mathbb{Z}\{v_1,v_2\} \rightarrow 0
$$
you can do purely algebraic identification $v:=v_1 = -v_2$, $\partial e = v_2 - v_1 = 2 v$ and get chain complex
$$
0 \overset{0}{\rightarrow} \mathbb{Z}\{e\} \overset{2}{\rightarrow} \mathbb{Z}\{v\} \overset{0}{\rightarrow} 0
$$
which is kind of wierd and I can't interpret it geometrically. You have one edge and it's border is twice the point.
One more thing about factoring chain complex if you have chain complex
$$
\dots \overset{\partial_{n+1}}{\rightarrow} C_n \overset{\partial_n}{\rightarrow} C_{n-1} \overset{\partial_{n-1}}{\rightarrow} \dots
$$
and subgroups $C'_{n} \unlhd C_n$. You need to make sure that $\partial_n(C'_n) \subset C'_{n-1}$ in order to get chain complex(actually you need it in order to be able properly define $\partial_n$ on those factor groups)
$$
\dots \overset{\partial_{n+1}}{\rightarrow} C_n/C'_n \overset{\partial_n}{\rightarrow} C_{n-1}/C'_{n-1} \overset{\partial_{n-1}}{\rightarrow} \dots
$$

So to fully answer you question I need to know how you identify faces of $\Delta^n$
