Accept $ X $ is a topological space and I want to talk about his "homology".

There is this notion of singular homology obtained from the singular chain complex. This is not very easy to calculate.

Suppose we assume that there is an additional structure in topological space $ X $So we can talk about the structure of a CW complex and from there to the concept of the cellular chain complex and cellular homology. This is easier than calculating a singular homology.

Let us continue with this topological space $ X $ (which we have assumed to have a CW structure) has an additional structure of a simplicial complex, then we can talk about the concept of the simplicial chain complex and then the concept of simplicial homology. This is easier to calculate than cellular homology.

Then it is the standard result that any two homology groups that come from different approaches coincide, if both are meaningful.

Question: Is there an extra structure (not trivial) that I can add to one?

Space with a simple structure that makes it easier to calculate

Homology in terms of chain complex easier than simple chain

Complex?

The same applies to cohomology. Suppose I have a topological space $ X $I can talk about his singular Cochain Complex and the corresponding singular cohomology.

Suppose this topological space $ X $ Given the structure of a manifold, we can talk about the Cochain complex of differential forms and use it to compute the cohomology of topological space $ X $, It is a standard result that if the coefficients are correct, the singular cohomology is identical to the deRham cohomology (deRham theorem).

Question: Is there an additional structure that I can add to a manifold?

this results in a simpler cochain complex than the cochain complex of

Differential forms that give a simpler way to calculate the cohomology of

the manifold? Suppose I fix a connection on the tangent

bundle up $ TM rightarrow M $ of the distributor $ M $ (or a Riemann metric on the manifold $ M $), I can produce easier

Complex with the compound that calculates cohomology easily?

When I try to find a meaning for the concept of cohomology theory, which connects to the tangent bundle $ TM rightarrow M $ (a metric on the distributor $ M $) then it is to be expected that this concept does not depend on the connection choice that I have defined. Does the assumption that there is a flat connection on the tangent bundle indicate an apparent cochain complex?

Suppose I ask that distributor $ M $ If a Lie group has an additional structure, there is a simpler Cochain complex that computes the cohomology of the manifold more easily than the DeRham cohomology. This is too much to ask, I am looking for results that lie between the deRham cohomology of manifoldness and the strictly lower structure than the notion of Lie groups.

References are welcome.