Let K be a simplicial complex. We introduce below boundary homomorphisms ∂_q: C_q(K) → C_q−1(K) between the chain groups of K. If σ is an oriented q-simplex of K then ∂_q(σ) is a (q − 1)-chain which is a formal sum of the (q − 1)-faces of σ, each with an orientation determined by the orientation of σ.

Let σ be a q-simplex with vertices v₀, v₁, . . . , v_q. For each integer j between 0 and q we denote by hv0, . . . , vˆ_j, . . . , vqi the oriented (q − 1)-face 〈v₀, . . . , v_j−1, v_j+1, . . . , v_q〉

of the simplex σ obtained on omitting v_j from the set of vertices of σ. In particular

             〈vˆ₀, v₁, . . . , v_q〉 ≡ 〈v₁, . . . , v_q〉, 〈v₀, . . . , v_q−1, vˆ_q〉 ≡ 〈v₀, . . . , v_q−1〉.

Similarly if j and k are integers between 0 and q, where j < k, we denote by

〈v₀, . . . , vˆ_j , . . . , vˆ_k, . . . v_q〉

the oriented (q−2)-face 〈v₀, . . . , v_j−1, v_j+1, . . . , v_k−1, v_k+1, . . . , v_q〉 of the simplex σ obtained on omitting vj and vk from the set of vertices of σ.

We now define a ‘boundary homomorphism’ ∂_q: C_q(K) → C_q−1(K) for each integer q. Define ∂_q = 0 if q ≤ 0 or q > dim K. (In this case one or other of the groups C_q(K) and C_q−1(K) is trivial.) Suppose then that 0 < q ≤ dim K. Given vertices v₀, v₁, . . . , v_q spanning a simplex of K, let

Inspection of this formula shows that α(v₀, v₁, . . . , v_q) changes sign whenever two adjacent vertices v_i−1 and vi are interchanged.

Suppose that v_j = vk for some j and k satisfying j < k. Then

α(v₀, v₁, . . . , v_q) = (−1)^j〈v₀, . . . , vˆ_j, . . . , v_q〉 + (−1)^k〈v₀, . . . , vˆ_k, . . . , v_q〉,

since the remaining terms in the expression defining α(v₀, v₁, . . . , v_q) con tain both v_j and v_k. However (v₀, . . . , vˆ_k, . . . , v_q) can be transformed to  (v₀, . . . , vˆ_j, . . . , v_q) by making k − j − 1 transpositions which interchange v_j successively with the vertices v_j+1, v_j+2, . . . , v_k−1. Therefore

〈v₀, . . . , vˆ_k, . . . , v_q〉= (−1)^k−j−1〈v₀, . . . , vˆ_j, . . . , v_q〉.

Thus α(v₀, v₁, . . . , v=) = 0 unless v₀, v₁, . . . , v_q are all distinct. It now follows immediately from Lemma 6.2 that there is a well-defined homomorphism ∂_q: C_q(K) → C_q−1(K), characterized by the property that

whenever v₀, v₁, . . . , v_q span a simplex of K.

Lemma 1.1 ∂_q−1 ◦ ∂_q = 0 for all integers q.

Proof The result is trivial if q < 2, since in this case ∂_q−1 = 0. Suppose that q ≥ 2. Let v₀, v₁, . . . , v_q be vertices spanning a simplex of K. Then