Question

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...)

Answer 1

A while back I taught a class on Morse theory where, among other things, I presented the Atiyah-Bott localization theorem in the special case of $S^1$. (I did not want to use more sophisticated commutative algebra.) In particular, the key argument in the proof is the Mayer-Vietoris theorem. For details, see Section 3.5 of these course notes. The Mayer-Vietoris theorem appears crucially in Lemma 3.65.

The original source (Atiyah & Bott's paper) is in my view still the best place to learn about this. Borel's Seminar on Transformation groups (precise reference in the above course notes) is also a source I found very helpful.

Answer 2

Regarding your last question: torsion is essential whenever the map $M^G \to M$ (inclusion of fixed points) fails to induce an isomorphism on equivariant cohomology (as opposed to isomorphism mod torsion). Probably the simplest example is given by the standard action of $S^1$ on $S^2$ with two fixed points. The inclusion of fixed points is the map $S^0 \to S^2$. The induced map on Borel constructions is equivalent to the map ${\mathbb C}P^\infty \coprod {\mathbb C}P^\infty \to {\mathbb C}P^\infty \vee {\mathbb C}P^\infty$, which is not a rational cohomology isomorphism.

Regarding your first question, Meyer-Vietoris sequence plays an important role in the proof, but it is not, I think, the entire proof. Basically, it is used to reduce the theorem to the following assertion: For a proper subgroup $H$ of $G$, the cohomology of $BH$ is a torsion module over $H^*(BG)$.