Let $E$ be a smooth oriented vector bundle over a smooth manifold $M$ and let $E^0$ be the complement of the zero section in $E$. I would like a reasonably explicit isomorphism between the relative De Rham cohomology group $H^p(E,E^0)$ and $H^p(M)$. Perhaps this is some version of the Thom isomorphism?
In case anyone needs a refresher, $H^*(E,E^0)$ is the cohomology of the complex whose chain groups are $\Omega^p(E,E^0) := \Omega^p(E) \oplus \Omega^{p-1}(E^0)$ and whose differential is given by $d(\omega_1, \omega_2) = (d \omega_1, i^* \omega_1 – d \omega_2)$.
EDIT: I actually want to prove that $H_{cv}^P(E,E^0) \cong H^p(M)$, where $H_{cv}^p(E,E_0)$ is the cohomology of the complex described above where all forms have compact support in the vertical direction.
Best Answer
I don't think that what you're asking is true exactly as stated, but what you're getting at is the Thom isomorphism. The proof of the isomorphism really only uses basic facts about vector bundles and axioms of homology, so pick any proof that you like and you should be able to translate it to relative de Rham cohomology. I won't prove it, but I can tell you explicitly what the maps are.
Given a vector bundle $p:E \to M$ of rank $r$, pick some metric on $E$ and use it to define the disc bundle $D \to M$ and unit sphere bundle $S \to M$. An orientation of $E$ gives us a class $\alpha \in H^r(D, S)$ that satisfies the property that for all $x \in M$, the restriction of $\alpha$ to the fiber above $x$ yields a generator of $H^r(D_x, S_x)$. Then we can define a map $H^k(M) \to H^{k+r}(D, S)$ via $\omega \mapsto p^\ast(\omega) \cup \alpha$. The Thom isomorphism theorem tells us that this map is an isomorphism of graded rings. If we use the de Rham model, then elements of $H^\ast(D, S)$ can be represented by differential forms, and the inverse map $H^{k+r}(D,S) \to H^k(M)$ is given by fiberwise integration.
Note that by homotopy invariance, $H^\ast(D, S) \cong H^\ast(E, E^o)$, but I prefer to use $(D,S)$ since it makes the fibers compact.
The best reference I can think of for the de Rham approach is Bott and Tu, Differential Forms in Algebraic Topology.