§ An invitation to homology and cohomology, Part 2 --- Cohomology
Once again, we have our humble triangle with vertices ,
edges , faces with boundary maps ,
We define a function on the vertices as:
We now learn how to extend this function to the higher dimensional objects,
the edges and the faces of the triangle.
To extend this function to the edges, we define a new function:
- , , .
Expanded out on the example, we evaluate as:
More conceptually, we have created an operator called (the coboundary operator)
which takes functions defined on vertices to functions defined on edges. This
uses the boundary map on the edges to "lift" a function on the vertices to a
function on the edges. It does so by assigning the "potential difference" of
the vertices to the edges.
We can repeat the construction we performed above, to construct another operator
in exactly the same way as we did before. For example, we can evaluate:
- , where
What we have is a chain:
Where we notice that , since the function that we have gotten
evaluates to zero on the face . We can prove this will happen in general,
for any choice of .
(it's a good exercise in definition chasing).
Introducing some terminology, A differential form is said to be a closed differential form
In our case, is closed, since . On the other hand
is not closed, since .
The intuition for why this is called "closed" is that its coboundary vanishes.
§ Exploring the structure of functions defined on the edges
Here, we try to understand what functions defined on the edges can look like,
and their relationship with the operator. We discover that there are
some functions which can be realised as the differential
of another function . The differential
forms such as which can be generated a through the operator
are called as exact differential forms. That is, exactly,
such that there is no "remainder term" on applying the operator.
We take an example of a differential form that is not exact, which has been
defined on the edges of the triangle above. Let's call it .
It is defined on the edges as:
We can calcuate the same way we had before:
Since , this form is not exact.
Let's also try to generate from a potential. We arbitrarily fix the
potential of to . That is, we fix , and we then try to
see what values we are forced to values of across the rest of the triangle.
Hence, there can exist no such such that .
The interesting thing is, when we started out by assigning ,
we could make local choices of potentials that seemed like they would fit
together, but they failed to fit globally throughout the triangle. This
failure of locally consistent choices to be globally consistent is
the essence of cohomology.
- . .
- . .
- . This is a contradiction!
- Ideally, we need for the values to work out.
§ Cohomology of half-filled butterfly
Here, we have vertices , edges
and faces .
Here, we see a differential form that is defined on the edges,
and also obeys the equation (Hence is closed). However, it
does not have an associated potential energy to derive it from. That is,
there cannot exist a certain such that .
So, while every exact form is closed, not every closed form is exact.
Hence, this that we have found is a non-trivial element of ,
since , hence , while there does not exist
a such that , hence it is not quotiented by the image of
So the failure of the space to be fully filled in (ie, the space has a hole),
is measured by the existence of a function that is closed but not exact!
This reveals a deep connection between homology and cohomology, which is
made explicit by the Universal Coefficient Theorem