Homology and Cohomology

In Homotopy Theory we discussed the notion of homeomorphism, homotopy, and homotopy equivalence, and gave intuitive notions of what ideas they are supposed to communicate. We also discussed what it means for a space to be path connected and simply connected, and the use of loops on a space to investigate certain properties of a space such as how many pieces it is composed of and if there exist “holes” on the space.

Loops are “deformations” of the circle; hence we have defined the set (which also happens to form a group) of equivalence classes of loops on the space $X$ “deformable” to each other as $[S^{1}, X]$ . Similarly, the other homotopy groups are defined as the set of equivalence classes $[S^{n}, X]$ , where $S^{n}$ is the n-dimensional sphere. In this post we will define another notion, that of a “cycle”, which also expresses ideas related to circles and more generally $n$ -dimensional spheres. Just as loops and their higher-dimensional counterparts play a central role in homotopy theory, cycles and the related concept of boundaries also play a central role in homology theory.

First we note that when we speak of circles, we do not usually include the interior. But we have a different term for the interior; we call it the open disk. The open disk and the circle together form the closed disk. Similarly when we speak of the sphere, we refer only to the surface of the sphere and not its interior. We call the interior the open ball, and the open ball and the sphere together the closed ball. This terminology also generalizes to $n$ -dimensional spheres as well. The interior of the $n$ -dimensional sphere is called the $n+1$ -dimensional open ball and both of them together form the $n+1$ -dimensional closed ball.

We note again that the $0$ -dimensional sphere of radius $R$ can be thought of just the two points $x=-R$ and $x=R$ on the real line. Its interior, the $1$ -dimensional open ball, is the set of all real numbers between $-R$ and $R$ , i.e. the set of all real numbers $x$ such that $-R<x<R$ , i.e. the open interval $(-R,R)$ . The $1$ -dimensional closed ball is then the closed interval $[-R,R]$ .

Intuitively, the $n$ -dimensional sphere is the boundary of the $n+1$ -dimensional closed ball (we will sometimes speak of just the boundary of a ball or a disk, hoping that this will cause no confusion). For example, the boundary of an interval is made up of its two endpoints. If we were to consider some other shape, like, say, a more general curve with endpoints, intuitively we could still think of these endpoints as forming the boundaries of the curve. However, some curves, such as the circle, or any closed loop, do not have endpoints, and therefore do not have a boundary. Shapes that have no boundary are called cycles.

We recall that we have been thinking of the circle itself as being the boundary of a disk. Combined with our observation that the circle does not have a boundary, this provides us with an example of the following important principle central to homology theory:

A shape which is the boundary of some other shape, has itself no boundary.

In other words:

All boundaries are cycles.

However, the converse is not actually true. Not all cycles are boundaries. Intuitively we think of circles as boundaries of disks because we have been subconsciously embedding them in the plane. We can come up with examples of circles which are not the boundaries of disks if we think of them as being parts of some other surface other than the plane. Still, this is probably quite confusing, so we will attempt to show what we mean by explicitly giving some examples.

But first we consider another space in which, like the plane, all circles are the boundaries of disks. We consider an ordinary sphere. One can think of, say, a basketball. We could take a pen and draw circles or loops on this basketball, and each circle or loop would bound some part of the basketball. If we take a pair of scissors and cut the basketball along the circle or loop that we have drawn, we will end up with a piece of rubber in the shape of the region bounded by the circle or loop. If we drew a circle, this region will be a disk. Hence, on a sphere, all circles are boundaries of disks.

Now let us consider an example of a surface in which not all circles are boundaries of disks. We consider the torus. It is the shape of a surface of a donut, but we can also think of the inner tube of a tire, which people also often use as flotation aids in swimming pools. We can still draw a circle bounding a disk on this surface, so that if we cut along the circle with a pair of scissors we still get a piece of rubber in the shape of a disk. However, we can also draw a circle around the “body” of the tube; if we cut along this circle, we would just cut the tube into something like a cylinder, since the circle was “bounding” no part of the tube, only the empty space inside (or it could have been filled with air).

There is another circle we can draw, around the “hole” in the middle of the inner tube, and if we cut it open, we just “open up” the inner tube. Once again this circle is not the boundary of a disk on the inner tube. This circle, along with the one we have considered earlier, still do not have any boundary, and yet, they are not boundaries of disks either. Therefore we see that on the torus, not all cycles are boundaries.

We see also that keeping track of whether there are cycles that are not boundaries give us some information about the space these cycles are on, the same way that keeping track of the loops that cannot be contracted to a point give us information about the space the loops are on.

To help formalize these ideas (although we won’t completely formalize them in this post), we note that the dimension of the boundary of a shape is one less than dimension of the shape itself. So, for example, let us consider a set of shapes of dimension $n$ , which we write as $C_{n}$ . We also have another set of shapes of dimension $n-1$ , which we write as $C_{n-1}$ . We now want the boundary of a shape in $C_{n}$ to be found in $C_{n-1}$ , and we want a “boundary function” that assigns to a shape in $C_{n}$ its boundary in $C_{n-1}$ . We write this boundary function as $\partial_{n}$ .

Some of the shapes in $C_{n-1}$ also have boundaries, and these boundaries are to be found in yet another set $C_{n-2}$ . The boundary function that sends shapes in $C_{n-1}$ to their boundaries in $C_{n-2}$ is written $\partial_{n-1}$ .

All these sets must have “zero elements” to allow for the case when a shape has no boundary. If a shape in $C_{n}$ has no boundary, then the boundary function sends it to the zero element in $C_{n-1}$ .

If we then define an abelian group structure on the sets $C_{n}$ , $C_{n-1}$ , and $C_{n-2}$ , with the zero element being the identity of the group, we can then define the cycles to be the kernel of the boundary function. Recall that the kernel of a function between groups is the subset of the domain that the function sends to the identity element in the range. We can also define the boundaries as the image of the boundary function. Recall that the image of a function is the subset of the range made up of the elements the function assigned to the elements of the domain.

Note that that the function obtained by composing the two successive boundary functions, $\partial_{n-1}\circ\partial_{n}$ , sends any element of $C_{n}$ to the identity element in $C_{n-2}$ . This is simply a reformulation of our “important principle” above which states that all boundaries are cycles.

We can now generalize the idea expressed by the groups $C_{n}$ , $C_{n-1}$ , and $C_{n-2}$ , so that we can have any number of groups indexed by the natural numbers, and boundary functions between two successive groups, which obey the property that the composition of two successive boundary functions will send any element of its domain to the identity element in its range. These groups together with the boundary functions between them form what is called a chain complex.

We can now define the homology groups. Since our shapes now form groups, we can use the law of composition of the group to define an equivalence relation between the elements of the group and form a quotient group (see also Groups and Modular Arithmetic and Quotient Sets). What we want is to declare two cycles in the group equivalent if they differ by a boundary. The $n$ -th homology group, written $H_{n}$ , is then defined as

$H_{n}=\text{Ker }\partial_{n}/\text{Im }\partial_{n+1}$ .

Here $\text{Ker }\partial_{n}$ refers to the kernel of the $n$ -th boundary operator, i.e. the cycles in $C_{n+1}$ and $\text{Im }\partial_{n}$ refers to the image of the $n+1$ -th boundary operator, i.e. the boundaries in $C_{n+1}$ . Recall that what we are doing is keeping track of the cycles that are not boundaries. We declare two cycles equivalent if they differ by a boundary, so any cycle which is also a boundary is declared equivalent to the identity element of the group, i.e. the zero element. If we write the law of composition of the group using the symbol “ $+$ “, we can express the equivalence relation as

$z+b\sim z$

where $z$ is a cycle and $b$ is a boundary. We can therefore easily see that

$0+b\sim 0$ .

This expresses the idea that what we are interested in are the cycles that are not boundaries. We are not so interested in the cycles that are boundaries, so we hide them away by declaring them to be equivalent to the identity element or zero element.

The sets of functions from the abelian groups that make up the chain complex to another abelian group form what is called a cochain complex of abelian groups, with its own coboundary functions. If we write the set of functions from $C_{n}$ to some other abelian group as $C_{n}^{*}$ , the coboundary function will go in the opposite direction as the boundary function. Whereas the boundary function $\partial_{n}$ sends elements from $C_{n}$ to their boundaries in $C_{n-1}$ , the coboundary function $d_{n-1}$ sends elements from $C_{n-1}^{*}$ to their coboundaries in $C_{n}^{*}$ . Note, once again, that while $C_{n}$ is a set of shapes (which happen to form an abelian group), $C_{n}^{*}$ is a set of functions from shapes to some other abelian group (which also happen to form an abelian group). The $n$ -th cohomology group, written $H^{n}$ is then defined as

$H^{n}=\text{Ker }d_{n}/\text{Im }d_{n-1}$ .

We have not yet explained how we are to define the shapes and abelian groups that make up our chain complex. We have relied only on the intuitive idea of cycles and boundaries. The methods by which these shapes and abelian groups are defined, such as singular homology and cellular homology, can be found in the references listed at the end of this post.

References:

Homology on Wikipedia

Cohomology on Wikipedia

Chain Complex in Wikipedia

Algebraic Topology by Allen Hatcher

A Concise Course in Algebraic Topology by J. P. May

Theories and Theorems

Math and Physics for Everyone

Homology and Cohomology

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