Compute 2 ( 2 (P 1 P 2 P 3 P 4 )) and show that

Question:

Compute ∂₂(∂₂(P₁P₂P₃P₄)) and show that it is 0, completing the proof of Theorem 41.9.

Data from Theorem 41.9

Let X be a simplicial complex, and let C_n(X) be then-chains of X for n = 0, 1, 2, 3. Then the composite homomorphism ∂_n-1∂_n mapping C_n(X) into C_n-2(X) maps everything into 0 for n = 1, 2, 3. That is, for each c ∈ C_n(X) we have ∂_n-1 (∂_n(c)) = 0. We use the notation "∂_n_-1∂_n= 0," or, more briefly, "∂² = 0."

Proof Since a homomorphism is completely determined by its values on generators, it is enough to check that for an n-simplex σ, we have ∂_n-₁ (∂_n(σ)) = 0. For n = 1 this is obvious, since ∂₀ maps everything into 0. For n = 2,