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An Introduction to Cohomology

In algebraic topology, homology has to do with the boundary operator ∂. The boundary of an m-simplex σm is roughly the set of (m-1)-simplexes which are faces of σm. In contrast, cohomology has to do with the co-boundary operator δ. The co-boundary of an m-simplex σm is roughly the set of (m+1)-simplexes that have σm as a face. Both the boundary operator and the co-boundary operator have the property that repeated applications produce the empty set: i.e.,

∂(∂S) = ∅
δ(δS) = ∅

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 σim for i=1 to αm then a chain C on the m-simplexes is of the form

C = Σciσm.

where the ci'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 = Σci(∂(σim))

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:

The boundary of the 2-simplex is

∂(v0v1v2) = v0v1 + v1v2 + v2v0

The boundary of the 1-simplex vivj is just the chain vi-vj. Thus the boundary of the boundary of the triangle reduces to

∂(∂(v0v1v2)) = v0-v1 + v1-v2 + v2-v0

The cancellation of vi with -vi 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.

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