The space $\Delta^n$ with all faces of the same dimension.

Question

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 $A$?

Answer 1

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 $$ 0 \rightarrow 0 \rightarrow \mathbb{Z}_3 \rightarrow \mathbb{Z} \rightarrow 0 $$

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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)

$$ \dots \overset{\partial_{n+1}}{\rightarrow} C_n/C'_n \overset{\partial_n}{\rightarrow} C_{n-1}/C'_{n-1} \overset{\partial_{n-1}}{\rightarrow} \dots $$

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