The term roughly used above refers to two ways in which the complete explanation is more complex than indicated. First, in the matter of an (m-1)-simplex being a face of an m-simplex one has to take into account the orientation of the (m-1)-simplex relative to the m-simplex. Second, homology and cohomology deal with chains on sets of simplexes. A chain on the m-simplexes is a vector with integer-valued components. If the m-simplexes for something like a polyhedron are denoted as σ_i^m for i=1 to α_m then a chain C on the m-simplexes is of the form

C = Σc_iσ^m.

where the c_i's are integers.

A set of simplexes is just a vector in which the components are either 0 or 1. The empty set is just the vector with all components equal to 0.

The boundary of such a chain is then

∂C = Σc_i(∂(σ_i^m))

Illustration of the Boundary
and Co-Boundary Operators

The simplest reasonable case is that of a triangle, a 2-simplex. The vertices of the triangle are labeled v₀, v₁, and v₂.

The various simplexes are given below:

• 0-simplexes: {v₀, v₁, v₂}
• 1-simplexes: {v₀v₁, v₁v₂, v₂v₀}
• 2-simplex: { v₀v₁v₂}

The boundary of the 2-simplex is

∂(v₀v₁v₂) = v₀v₁ + v₁v₂ + v₂v₀

The boundary of the 1-simplex v_iv_j is just the chain v_i-v_j. Thus the boundary of the boundary of the triangle reduces to

∂(∂(v₀v₁v₂)) = v₀-v₁ + v₁-v₂ + v₂-v₀

The cancellation of v_i with -v_i for all i gives

∂(∂(v₀v₁v₂)) = 0

Thus the boundary of the boundary of a triangle is the empty set ∅.

Now consider the co-bounary of a vertex, say v₀,

δ(v₀) = v₀v₁ - v₂v₀

where the signs take into account that v₀ is the face of v₀v₁ where v₀v₁ is leaving v₀ whereas for v₂v₀ it is entering v₀.

The co-boundary of v₀v₁ is just the 2-simplex v₀v₁v₂. The co-boundary of v₂v₀ is the 2-simplex v₂v₀v₁, which is the same as v₀v₁v₂. Thus the co-boundary of the co-boundary of v₀ is given by

δ(δ(v₀)) = v₀v₁v₂ - v₀v₁v₂ = 0

Thus, at least in this case, the co-boundary of the co-boundary of a simplex is the empty set.

Cohomology differs from homology in that there may be an infinity of simplexes for which a given simplex is a face. Chains in cohomology are such that only a finite number of the components are non-zero.

Homology and cohomology are of course much more complex than this illustration indicates but the illustration does indicate the nature of boundaries and co-boundaries and why repeated applications of these operators produce the empty set.