appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley USA 


In algebraic topology, homology has to do with the boundary operator ∂. The boundary of an msimplex σ^{m} is roughly the set of (m1)simplexes which are faces of σ^{m}. In contrast, cohomology has to do with the coboundary operator δ. The coboundary of an msimplex σ^{m} is roughly the set of (m+1)simplexes that have σ^{m} as a face. Both the boundary operator and the coboundary operator have the property that repeated applications produce the empty set: i.e.,
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 (m1)simplex being a face of an msimplex one has to take into account the orientation of the (m1)simplex relative to the msimplex. Second, homology and cohomology deal with chains on sets of simplexes. A chain on the msimplexes is a vector with integervalued components. If the msimplexes for something like a polyhedron are denoted as σ_{i}^{m} for i=1 to α_{m} then a chain C on the msimplexes is of the form
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
The simplest reasonable case is that of a triangle, a 2simplex. The vertices of the triangle are labeled v_{0}, v_{1}, and v_{2}.
The various simplexes are given below:
The boundary of the 2simplex is
The boundary of the 1simplex v_{i}v_{j} is just the chain v_{i}v_{j}. Thus the boundary of the boundary of the triangle reduces to
The cancellation of v_{i} with v_{i} for all i gives
Thus the boundary of the boundary of a triangle is the empty set ∅.
Now consider the cobounary of a vertex, say v_{0},
where the signs take into account that v_{0} is the face of v_{0}v_{1} where v_{0}v_{1} is leaving v_{0} whereas for v_{2}v_{0} it is entering v_{0}.
The coboundary of v_{0}v_{1} is just the 2simplex v_{0}v_{1}v_{2}. The coboundary of v_{2}v_{0} is the 2simplex v_{2}v_{0}v_{1}, which is the same as v_{0}v_{1}v_{2}. Thus the coboundary of the coboundary of v_{0} is given by
Thus, at least in this case, the coboundary of the coboundary 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 nonzero.
Homology and cohomology are of course much more complex than this illustration indicates but the illustration does indicate the nature of boundaries and coboundaries and why repeated applications of these operators produce the empty set.