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I've just started to study homology theory. And I'm trying to calculate all $H_n(\Delta_N)$ for some $N$. I know that the number of $m$-simplex in $N$-simlex is $b_{N,m}={N+1 \choose m+1}=\frac{(N+1)!}{(m+1)!(N-m)!}$.

So $C_0=\mathbb Z^{N+1}$, $C_1=\mathbb Z^{\frac{N(N+1)}{2}}$, ... , $C_N=\mathbb{Z}$, $C_{N+1}=0$, ...

And here I have some misunderstanding. As I know I shoud calculate $H_n(\Delta_N)=Ker(\partial_n)/Im(\partial_{n+1}))$. In opposite to standard examples I don't know what I should do next. Maybe there is some another ways to solve this problem?

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It's less important to count the number of $m$-simplices and more important to know where the boundary maps $C_{m+1} \to C_m$ send them! Perhaps if you know what the answer is it will be easier to see what you need to do: so as a hint, I'll tell you that $H_0 \cong \mathbb{Z}$ and $H_n = 0$ for $n > 0$. –  Zhen Lin Sep 17 '13 at 16:49
@Zhen Lin Thanks for your comment! I understand why $Ker(\partial_0)=\mathbb{Z^{n+1}}$, but I don't get why $Im(\partial_1)=\mathbb{Z^{n}}$. –  Oiale Sep 17 '13 at 18:34
Use the fact that homology groups are homotopy invariant. Then compute homology of a point. –  studiosus Sep 17 '13 at 20:25

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