MathOverflow

In a previous question regarding bitangent circles, I realized that the underlying geometric mechanism might point to a much broader topological phenomenon. To avoid confusion with specific cases like circles or conics, I would like to propose and discuss...

Let $X,Y$ be $G$ spaces, where $G$ is a compact Lie group. Is there a Kunneth (isomorphism) formula for Bredon cohomology with the constant rational Mackey functor $\underline{\mathbb{Q}}$ given by$$H^*_{G}(X \times Y; \underline{\mathbb{Q}}) \cong H...

In short: can someone explain the argument being alluded to in SGA 7, Expose XIV (Comparaison avec la theorie transcendante), Section 1.3.6?
In more detail: the setup is:
$D$ the open unit disk in $\mathbb{C}$

Let $k$ be a field and $R$ be the ring $k[x,xy,xy^2,xy^3]$. Let $X$ be $\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of $X$ along the maximal ideal $I$ of $R$ generated by $x,xy,xy^2,xy^3$.
I have proved that $\tilde{X}$ is a line bundle over $\mathbb{P}^1_k$, so that the $G$-theory groups of $\tilde{X}$ agree with the $G$-theory groups of $\mathbb{P}^1_k$....