# (elementary?) proof of simplicial homology groups of $\Delta_n$

Using the (trivial) CW-complex structure of $$\Delta_n$$, I would like to compute the homology groups of $$\Delta_n$$. It's obvious that (for $$k \leq n$$) $$C_k \simeq \mathbb{Z}^{{n+1 \choose k+1}}$$, and that $$im \delta_{k+1} \subset ker \delta_k$$. It's the reverse inclusion for $$k \geq 1$$ that I'm not able to show.

I know how to prove this using homotopy invariance of homology, as $$\Delta_n$$ is contractible, and a proof in that vein is given in a comment here Explicit calculation of simplicial homology - but I'm wondering if there is something more elementary? The CW-complex structure makes computing the homology groups of the sphere almost trivial, and I was hoping that it would be easy for the disk as well.

However, if you wish to sweep these things under the rug: the n-simplex is homeomorphic to the n-disk which has a cell structure given by a n-cell attached along the identity to the (n-1)-sphere. The image of the n-cell under the boundary map is the (n-1)-cell, so the n-1 homology is trivial (unless n-1=0 when degrees are tricky to define). The n homology is also trivial because the boundary map is an isomorphism here. The 0 homology is $$\mathbb{Z}$$ since the boundary map into it is $$0$$ (if $$n>1$$ because there are no 1-cells and if $$n=1$$ because the orientations of the boundary points of the circle cancel each other out).
It's quite easy in simplicial homology to write down a contracting homotopy for the identity map of $$C(\Delta_n)$$. Let the vertices of $$\Delta_n$$ be labelled $$0,1,\ldots,n$$. Then the $$k$$-chains are labelled by tuples $$(j_0,\ldots,j_k)$$ where $$0\le j_0. Define $$T:C_k(\Delta_n)\to C_{k+1}(\Delta_n)$$ by $$T:(j_0,\ldots,j_k)\mapsto(0,j_0,\ldots,j_k)$$ if $$j_0\ge1$$ and $$T:(j_0,\ldots,j_k)\mapsto0$$ if $$j_0=0$$. If then $$\delta_k(\gamma)=0$$ for $$\gamma\in C_k$$ (with $$k\ge1$$) then $$\gamma=\pm\delta_{k+1}(T(\gamma))$$ (I can't be bothered to check the sign), so $$\ker\delta_k\subseteq\mathrm{im}\,\delta_{k+1}$$.