We have this theorem: Let $U$, $V$ two open sets of manifold $M$, ($U \cup V = M$). If they are $G$ -stable the induced sequence in cohomology

$$ \cdots \to H^k_G (U \cup V) \to H^k_{G}(U)\oplus H^k_G(V) \to H^k_{G}(U \cap V) \to H^{k+1}_{G}(U \cup V) \to \cdots $$

is exact. There is a Borel localization theorem: Let $M$ a compact manifold equipped with a $G$- action ($G$ is a compact Lie group). Let $i:F \rightarrow M$ denote the inclusion of the $G$-fixed point set of $M$ in $M$ of the set of $M$. Then

$$ i^{*}: H^\bullet_G(M) \to H^\bullet_G(F) \simeq H^\bullet(F) \otimes H^\bullet_{G}(pt) $$ is an isomorfism modulo $H^{*}_{pt}(G)$-torsion.

Is there a way to give a proof of Borel localization theorem using equivariant Mayer-Vietoris theorem? Is there a good (concrete) example in which the torsion is essential to have isomorfism? (when $H^\bullet_{G}(pt)$ is a polinomial ring...)