Homology, An Introduction
Introduction
Let G1 G2 G3 etc be a sequence of abelian groups,
with group homomorphisms
f1(G1) into G2,
f2(G2) into G3,
f3(G3) into G4, and so on.
Furthermore, the image of each homomorphism lies in the kernel of the next homomorphism.
Thus the composition of two successive homomorphisms is always 0.
Such a sequence, with images mapping into kernels, is called a chain, or a chain complex.
This is somewhat unfortunate, because in other contexts
a chain of groups is nested, one inside another.
We talk about ascending and descending chains of groups etc.
Well this is a different kind of chain, and perhaps that is why it is sometimes called a chain complex.
The internal homomorphisms are called boundary operators,
a term that is associated with algebraic topology.
The homology group Hi is the kernel of fi mod the image of fi-1.
Remember, we said each image lies inside the subsequent kernel, so these homology groups are well defined.
You may be wondering what H1 is, since there is a kernel of f1,
but no prior group, and no image.
If the chain has a left end, the homology group is not defined at that point.
similarly, if the chain has a right end, the homology group is not defined their either.
To avoid this, the chain may run forever in both directions.
Thus there is a G0, G-1, G-2, and so on.
Let's look at a silly example.
Let a chain complex run forever in both directions.
Let every group in the chain be Z7,
the integers mod 7.
Let every homomorphism squash Z7 down to 0 in the next group.
The image is always 0, and the kernel is always all of Z7.
In each case the homology group is Z7.
Modules
Given a ring R, the groups in a chain could be replaced with modules,
withmodule homomorphisms carrying each into the next.
Again, the image of each homomorphism must lie in the kernel of the following homomorphism.
The quotient (kernel mod image) gives the homology module.
Since abelian groups are Z modules,
I will usually talk about modules,
whence the results automatically apply to abelian groups.
Notation
In a chain of modules, arrows indicate the homomorphisms.
Sometimes they are labeled, if we want to refer to the homomorphisms individually.
Here is an example with generic arrows.
Notice that I've switched to the letter M, for module.
M0 → M1 → M2 → M3 → …
Reverse Convention
Sometimes the boundary operators run in reverse.
That is, Mi maps into Mi-1.
In this case the arrows run to the left, or,
the indices decrease as you move to the right.
Personally I prefer the convention shown above, and that's what I will use throughout this topic.
But bear in mind, algebraic topology uses the reverse convention, for good reason;
the groups actually map into groups that exist at a lower dimension, thus a lower index.