I would like to know whether the following co boundary or boundary maps can introduce some new kinds of cohomology or homology which contains some non trivial, new, finite dimensional and helpful information about the manifold or the dynamics of a vector field on it.
All maps $\phi$ described below are defined on the chain of differential forms on a manifold $M$ which obviously satisfy $\phi \circ \phi=0$.
In this question, to sake finite dimensionality of resulting cohomology or homology, instead of consideration of the whole complex $\Omega^*(M)$, we are flexible to consider an appropriate sub complex of $\Omega^*(M)$. Each operator $\phi$ listed below can suggest some subcomplex as domain of definition of $\phi$.
Let $M$ be a Riemannian manifold with Laplace operator $\Delta$ and $X$ be a vector field on $M$.
Our differential maps(coboundary or boundary maps) are the following maps. The first $2$ maps depends on the Riemannian structure and the last $2$ maps, which decrease the degree by 2, depend on the vector fields $X$.
Suggestion for $\phi:$
The map $\phi$ is:
1)$d\circ \Delta$
2)$d^* \circ \Delta$
3)$d^*\circ L_X \circ d^*$
4)$i_X\circ \Delta \circ i_X$
The operator $L_X$ is the Lie derivative operator. The operator $i_X:\Omega^i(M) \to \Omega^{i-1}(M)$ is the interior product and $d^*$ is the codifferential operator.