Let K be a simplicial complex. For each non-negative integer q, let ∆_q(K)  be the additive group consisting of all formal sums of the form

where n₁, n₂, . . . , ns are integers and v^r₀, v^r₁, . . . , v^r_q are (not necessarily distinct) vertices of K that span a simplex of K for r = 1, 2, . . . , s. (In more formal language, the group ∆_q(K) is the free Abelian group generated by the set of all (q + 1)-tuples of the form (v₀, v₁, . . . , v_q), where v₀, v₁, . . . , v_q span a simplex of K.)

We recall some basic facts concerning permutations. A permutation of a set S is a bijection mapping S onto itself. The set of all permutations of some set S is a group; the group multiplication corresponds to composition of permutations. A transposition is a permutation of a set S which interchanges two elements of S, leaving the remaining elements of the set fixed. If S is finite and has more than one element then any permutation of S can be expressed as a product of transpositions. In particular any permutation of the set {0, 1, . . . , q} can be expressed as a product of transpositions (j −1, j)  that interchange j − 1 and j for some j.

Associated to any permutation π of a finite set S is a number ℰ_π, known as the parity or signature of the permutation, which can take on the values ±1.

If π can be expressed as the product of an even number of transpositions,  then ℰ_π = +1; if π can be expressed as the product of an odd number of transpositions then ℰ_π = −1. The function π → π is a homomorphism from the group of permutations of a finite set S to the multiplicative group {+1, −1} (i.e., ℰ_πρ = ℰ_πℰ_ρ for all permutations π and ρ of the set S). Note in particular that the parity of any transposition is −1.

Definition The qth chain group C_q(K) of the simplicial complex K is de fined to be the quotient group ∆_q(K)/∆⁰_q (K), where ∆⁰_q (K) is the sub group of ∆_q(K) generated by elements of the form (v₀, v₁, . . . , v_q) where v₀, v₁, . . . , v_q are not all distinct, and by elements of the form

                              (v_π(0), v_π(1), . . . , v_π(q)) − ℰ_π(v₀, v₁, . . . , v_q)

where π is some permutation of {0, 1, . . . , q} with parity ℰ_π. For convenience,  we define C_q(K) = {0} when q < 0 or q > dim K, where dim K is the dimension of the simplicial complex K. An element of the chain group C_q(K)  is referred to as q-chain of the simplicial complex K.

We denote by 〈v₀, v₁, . . . , v_q〉the element ∆⁰_q (K) + (v₀, v₁, . . . , v_q) of C_q(K) corresponding to (v₀, v₁, . . . , v_q). The following results follow immediately from the definition of C_q(K).

Lemma 1.1 Let v₀, v₁, . . . , v_q be vertices of a simplicial complex K that span a simplex of K. Then

• 〈v0, v1, . . . , vq〉 = 0 if v₀, v₁, . . . , vq are not all distinct,

• 〈v_π(0), v_π(1), . . . , vπ(q)〉 = ℰ_π〈v₀, v₁, . . . , v_q〉 for any permutation π of the set {0, 1, . . . , q}.

Example If v0 and v1 are the endpoints of some line segment then 〈v₀, v₁〉 = −〈v₁, v₀〉.

If v₀, v₁ and v₂ are the vertices of a triangle in some Euclidean space then

〈v₀, v₁, v₂〉= 〈v₁, v₂, v₀〉 = 〈v₂, v₀, v₁〉 = −〈v₂, v₁, v₀〉

                     = −〈v₀, v₂, v₁〉= −〈v₁, v₀, v₂〉.

Definition An oriented q-simplex is an element of the chain group C_q(K) of the form ±〈v₀, v₁, . . . , v_q〉, where v₀, v₁, . . . , v_q are distinct and span a simplex of K.

An oriented simplex of K can be thought of as consisting of a simplex of K (namely the simplex spanned by the prescribed vertices), together with one of two possible ‘orientations’ on that simplex. Any ordering of the vertices determines an orientation of the simplex; any even permutation of the ordering of the vertices preserves the orientation on the simplex, whereas any odd permutation of this ordering reverses orientation.

Any q-chain of a simplicial complex K can be expressed as a sum of the form

                   n₁σ₁ + n₂σ₂ + · · · + n_sσ_s

where n₁, n₂, . . . , ns are integers and σ₁, σ₂, . . . , σ_s are oriented q-simplices of K. If we reverse the orientation on one of these simplices σi then this reverses the sign of the corresponding coefficient n_i . If σ₁, σ₂, . . . , σ_s represent distinct simplices of K then the coefficients n₁, n₂, . . . , n_s are uniquely determined.

Example Let v₀, v₁ and v₂ be the vertices of a triangle in some Euclidean space. Let K be the simplicial complex consisting of this triangle, together with its edges and vertices. Every 0-chain of K can be expressed uniquely in the form

                             n₀〈v₀〉 + n₁〈v₁〉 + n₂〈v₂〉

for some n₀, n₁, n₂ ∈ Z. Similarly any 1-chain of K can be expressed uniquely in the form

              m₀〈v₁, v₂〉 + m₁〈v₂, v₀〉 + m₂〈v₀, v₁〉

for some m₀, m₁, m₂ ∈ Z, and any 2-chain of K can be expressed uniquely as n〈v₀, v₁, v₂〉 for some integer n

Lemma 1.2 Let K be a simplicial complex, and let A be an additive group.

Suppose that, to each (q + 1)-tuple (v₀, v₁, . . . , v_q) of vertices spanning a simplex of K, there corresponds an element α(v₀, v₁, . . . , v_q) of A, where

• α(v₀, v₁, . . . , v_q) = 0 unless v₀, v₁, . . . , v_q are all distinct,

• α(v₀, v₁, . . . , v_q) changes sign on interchanging any two adjacent vertices v_j−1 and v_j .

Then there exists a well-defined homomorphism from C_q(K) to A which sends 〈v₀, v₁, . . . , v_q〉 to α(v₀, v₁, . . . , v_q) whenever v₀, v₁, . . . , v_q span a simplex of K. This homomorphism is uniquely determined.

Proof The given function defined on (q + 1)-tuples of vertices of K extends to a well-defined homomorphism α: ∆_q(K) → A given by

for all permutations π of {0, 1, . . . , q}, since the permutation π can be ex pressed as a product of transpositions (j − 1, j) that interchange j − 1 with j for some j and leave the rest of the set fixed, and the parity επ of π is given by ε_π = +1 when the number of such transpositions is even, and by ε_π = −1 when the number of such transpositions is odd. Thus the generators of ∆⁰_q  (K) are contained in ker α, and hence ∆⁰_q (K) ⊂ ker α. The required homomorphism α˜: Cq(K) → A is then defined by the formula

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