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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 $$ \cdots \rightarrow H^{k}_{G}(U \cup V) \rightarrow H^{k}_{G}(U) \oplus H^{k}_{G}(V) \rightarrow H^{k}_{G}(U \cap V) \rightarrow H^{k+1}_{G}(U \cup V) \rightarrow \cdots $$ 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 $$ i^{*}: H^{*}_{G}(M) \rightarrow H^{*}_{G}(F) \simeq H^{*}(F) \otimes H^{*}_{G}(pt) $$ 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 $H^{*}_{G}(pt)$ is a polinomial ring...)

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