First, we make the following observation: let $X: M \rightarrow TM $ be a vector field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e. $$ i_X \circ i_{X} =0.$$ Is there any "name" for the corresponding "homology" group that one can define (Kernel mod image)? Has this "homology" group been studied by others (there are plenty of questions that one can ask........is it isomorphic to anything more familiar etc etc).

Similarly, a dual observation is as follows: Let $\alpha$ be a one form; taking the wedge product with $\alpha$ twice gives us zero. One can again define kernel mod image. Does that give anything "interesting"?

If people have investigated these questions, I would like to know a few references.

My purpose for asking the "name" of the (co)homology group is so that I can make a google search using the name. I was unable to do that, since I do not know of any key words under this topic (or if at all it is a topic).