# Why are contact structures studied from a cohomological, rather than homological, perspective?

As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a "homological" version using either contact vector fields or Reeb vector fields. Is there a good reason for this? Thanks.

From the distribution point of view, you would take a hyperplane subbundle $H$ in $TM$, and consider the quotient bundle $Q:=TM/H$ with projection $q:TM\to Q$. Then for vector fields $\xi$ and $\eta$ on $M$ which have values in $H$, you consider the projection $q([\xi,\eta])$ of their Lie bracket. One immediately verifies that the resulting map is bilinear of smooth functions, so there is an induced bundle map $\mathcal L:H\times H\to Q$. The condition that $H$ is a contact distribution then is that $\mathcal L$ is non-degenerate (as a bilinear form) in each point. This can also be phrased as the fact that $\mathcal L$ in each point $x$ makes $H_x\oplus Q_x$ into a Heisenberg-algebra.
$\kappa\wedge d\kappa\wedge\dots\wedge d\kappa\neq0$ is such a neat expression. When you are trying to express it in terms of tangent vector fields, it would be a mess. By the way, the Reeb field is frequently used in contact geometry.