The Thom space $T(E)$ of a vector bundle $E \to B$ with metric is defined as $D(E)/S(E)$, where $D(E)$ denotes the disk bundle and $S(E)$ denotes the sphere bundle of $E \to B$.
I've been trying to prove that $H^{*}(T(E))$ is isomorphic to $H^{*}(E,E_{0})$, where $E_{0}$ is the complement in $E$ of the zero section. I already proved that $H^{*}(E,E_{0})$ is isomorphic to $H^{*}(D(E),S(E))$ (using the long exact sequence of a pair and that $D(E)$ is a deformation retract of $E$ such that if $r$ is such deformation retraction then $r(E_0) \subset S(E)$.)
I think I have to use excision to the pair $(D(E), S(E))$ in order to realize the cohomology $H^{*}(D(E),S(E))$ as the cohomology $H^{*}(T(E))$ but I can't realize how to do it. Any suggestion?
Thanks in advance!
You did the hardest part already. By dualizing Proposition 2.22 in Hatcher or referring to Relative Cohomology Isomorphic to Cohomology of Quotient we get that $$ H^*(D(E),S(E)) = \tilde H^*(D(E)/S(E)) = \tilde H^*(T(E)).$$
Note that to deduce this we would need the pair to be a good pair, but this will be easy for you to show (if not completely obvious by what you already did). You need to show that $S(E)\subset D(E)$ is nonempty and closed and that there exists a neighborhood which deformationretracts onto it.