In short, a (co)homology theory is a way of assigning sets of algebraic invariants to classes of topological spaces by associating (co)chain complexes to a topological space and computing their (co)homology groups.

The de Rham cohomology is the specific cohomology theory associating to a smooth manifold the cochain complex of vector spaces of differential n-forms on the manifold with exterior derivative boundary maps.

In this series of posts, I’ll attempt to give a step-by-step explanation of what these two definitions actually mean, starting by building up the definition of a (co)homology theory and (co)chain complexes, then constructing the de Rham cohomology theory from scratch as an example, running through differential forms and the exterior derivative.

In this explanation, I will assume basic familiarity with topological spaces and smooth manifolds. If you don’t know the technical definitions of these words, you can check out my last series of posts on the mathematical definition of spacetime, where I run through and construct these definitions from scratch. These can be found on my blog under the tags #spacetime and #differential geometry.

I also assume ability to read and understand commutative diagrams, as well as familiarity with basic terms from group theory and linear algebra such as “kernel” or “image” and contextual use of the word “trivial”.

(Co)chain Complexes

(Co)chain complexes are very important objects for understanding (co)homology theories so we’ll start with them.

A chain complex is a sequence Aₙ of abelian groups paired with a sequence dₙ of group homomorphisms causing the following diagram to commute:

Similarly, a cochain complex is a sequence Aⁿ of abelian groups paired with a sequence dⁿ of group homomorphisms causing the following diagram to commute:

The homomorphisms labelled 0 in these diagrams are trivial homomorphisms mapping every element of their domain to the identity element of their codomain.

Commutativity of these diagrams therefore expresses that the image of a map in the sequence of group homomorphisms is a subset of the kernel of the next map in the sequence.

Each member of the sequences of abelian groups is called a (co)chain.

Members of the sequence of group homomorphisms are often referred to as (co)boundary operators.

For each n, Aₙ (or Aⁿ) is said to be of degree/dimension n.

It is worth noting here that the only difference between chain and cochain complexes is that boundary operators of chain complexes decrease the degree of a chain while coboundary operatoes of cochain complexes increase the degree of a cochain.

(Co)homology of a (Co)chain Complex

Given a chain complex (Aₙ, dₙ) we can construct its homology groups.

In particular, for any n, ker(dₙ) is a normal subgroup of Aₙ, and by the definition of the boundary operators, im(dₙ₊₁) ⊆ ker(dₙ).

Since ker(dₙ) is a normal subgroup, any general subgroup of ker(dₙ) is normal in ker(dₙ), hence we have that im(dₙ₊₁) ⊴ ker(dₙ).

This means we can compute the quotient group

Hₙ = ker(dₙ) / im(dₙ₊₁)

This quotient group Hₙ is exactly the nth homology group of the chain complex (Aₙ, dₙ).

In a similar fashion, the nth cohomology group of a cochain complex (Aⁿ, dⁿ) is defined as the quotient

Hⁿ = ker(dⁿ) / im(dⁿ⁻¹)

(Co)homology groups serve as important invariants of (co)chain complexes. Algebraic topologists use this fact by associating (co)chain complexes to (classes of) topological spaces, allowing them to use (co)chain (co)homology as an invariant for topological spaces!

The particular way in which a complex is chosen for each topological space is called a (co)homology theory.

These theories must usually also satisfy the Eilenberg-Steenrod axioms.

This may all seem fairly abstract and confusing right now, but fear not!

In the next part(s), we’ll discuss the Eilenberg-Steenrod axioms and run through de Rham cohomology as an example of a cohomology theory. I’ve chosen this example over the more commonly exemplar singular homology because it’s my post and I want to. Also de Rham is naturally isomorphic to singular cohomology with real coefficients by de Rham’s theorem, which is close enough for me.