# Simplicial Homology of Simplex without Deformations

Let $$\Delta$$ be the $$n-$$simplex with vertices $$1,\dots,n$$. I want to prove that the reduced homology of $$\Delta$$ is trivial without making an explicit mention to the fact that $$\Delta$$ is contractible.

I can use the fact that chain-homotopic chain complex maps induce the same homomorphism in homology, so I just need to find such maps and such chain-homotopy.

For a fixed vertex $$x$$ of $$\Delta$$, I've defined the inclusion map $$C(\{x\})\to C(\Delta)$$ and the candidate for its homotopical inverse is just the constant map which maps every vertex onto $$x$$. This way, the chain map is the inclusion in $$C_0$$, and it's $$0$$ in $$C_i$$ for $$i>0$$.

The composition map $$q:C(\Delta)\to C(\Delta)$$ is a chain map defined as follows: $$q_0$$ is induced by the constant map $$k\mapsto x$$ for every vertex $$k$$ of $$\Delta$$, and $$q_i=0$$ for $$i\geq 1$$.

I want to prove that $$q$$ is chain-homotopic to the identity map, so that it's proven that $$C(\{x\}),C(\Delta)$$ are chain-homotopically equivalent and they have the same homology. But I'm having a bit of trouble defining the chain homotopy $$P:C_i(\Delta)\to C_{i+1}(\Delta)$$.

Similarly to the proof of 2.10 in Hatcher's book, page 112, I want to define such a prism operator, but I get a bit confused since now I am in simplicial homology and I don't know how would I translate the homotopy into the corresponding part of the prism operator in simplicial homology.

In simplicial homology one constructs a contracting homotopy as follows. For a simplex $$\sigma$$ with vertices $$i_0 define $$P(\sigma)=\tau$$ with vertices $$1, when $$i_0>1$$ and $$P(\sigma)=0$$ otherwise. This is a homotopy between the identity map on $$C(\Delta)$$ and the simplicial map collapsing $$\Delta$$ onto the vertex $$1$$.