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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.,
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
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
The simplest reasonable case is that of a triangle, a 2-simplex. The vertices
of the triangle are labeled v0, v1, and v2.
∂(∂S) = ∅
δ(δS) = ∅ C = Σciσm.
∂C = Σci(∂(σim))
Illustration of the Boundary
and Co-Boundary Operators
The various simplexes are given below:
The boundary of the 2-simplex is
The boundary of the 1-simplex vivj is just the chain vi-vj. Thus the boundary of the boundary of the triangle reduces to
The cancellation of vi with -vi for all i gives
Thus the boundary of the boundary of a triangle is the empty set ∅.
Now consider the co-bounary of a vertex, say v0,
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
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.