I'm currently reviewing the Thom isomorphism and the de Rham cohomology of vector bundles over a compact manifold $M$.
I'm familiar with the case of topological disk bundles, where the Thom class of a bundle $E\to M$ is an element of $H^k(E,E^0)$, where $k$ is the dimension of the fiber $F \cong D^k$ and $E^0 $ is the complement of the zero section.
While going through Bott-Tu's Differential Forms in Algebraic Topology, one can find the Thom class of a vector bundle of rank $k$ defined as an element $\Phi(E)$ in $H^k_{cv}(E)$, where $H^*_{cv}(E)$ is the compact vertical cohomology of $E$. Bott and Tu later introduce relative de Rham cohomology, and point out that (as in the topological case) with that definition, one can consider the Thom class as an element of $H^k(E, E^0)$. The definition and some discussion on it can be found here.
However, their definition of relative cohomology doesn't seem quite transparent to me. For example, it is very different from the definition of relative singular cohomology.
An alternative definition can be found in Godbillon's Elements de Topologie Algebrique (sadly, I found no English translation), which is more similar to the one in the singular context: He defines $H^*(X,Y)$ as the cohomology of the complex $\Omega^*(X,Y)$ of forms in $X$ which vanish when pulled back to $Y$. The only disadvantage of this definition, as far as I can see, is that you need $Y$ to be embedded as a closed subset if you want to have homotopy invariance (for example, one has $\{0 \} = H^n(\mathbb{R}^n, \mathbb{R}^n - \{0 \}) \ncong H^n(\mathbb{R}^n, B(0,1)^c) \cong \mathbb{R})$.
My questions are the following:
Are there any advantages of introducing compact vertical cohomology? The Thom class could be defined as an element of $H^k(E,E')$, where $E'= \{ v \in E : || v|| \geq 1 \}$ if we use Godbillon's definition of relative cohomology, and (it seems to me) we would get the same results.
I haven't found that definition elsewhere though, so perhaps there's a reason it's fallen out of favor? It seems (to me) to be more clear than the one in Bott-Tu, and it also follows much more closely the definition in the singular case (besides the condition of having the subspace embedded as a closed set).
I'm in the middle of writing some notes on this, and I'm tempted to make the switch from $H^*_{cv}(E)$ to $H^*(E, E')$ since later I plan on going into the topological case, and I'd prefer to have the definitions be more similar to each other (perhaps the answers to this question will lead me to reconsider).
All comments and answers welcome! Thanks in advance.