Suppose that $G=S^1$ acts on a smooth, connected, compact manifold with discrete fixed points, additionally assume that there is at least one fixed point. 


Let $\alpha \in H^{2}\_{S^1}(M)$ be such that $\alpha|\_{p} = 0$ for any fixed point $p$. The space of such $\alpha$ forms an additive sub-group $A \subset H^{2}\_{S^{1}}(M)$. 


**Question**: 
is there an example where $A \neq \{0\}$?


How about if we assume that $M$ is a compact symplectic manifold with a Hamiltonian $S^1$-action with discrete fixed points?



Equivariant cohomology defined by restrictions?
This subgroup $A$ is precisely the torsion subgroup in $H^2\_{S^1}(M)$ (since $H^2\_{S^1}(M^{S^1})$ is obviously free, and the relative $H^2\_{S^1}(M,M^{S^1})$ is torsion). So, it will be trivial if and only if the equivariant cohomology in degree 2 is free. This is true for compact $M$ with Hamiltonian $S^1$-action by Kirwan (see, for example, Theorem 14.1 of <http://www.math.ias.edu/~goresky/pdf/equivariant.jour.pdf>), but I think this is too much to hope for in general (I don't know a connected counterxample off-hand; obviously, you can take a free action on a compact manifold, and take disjoint union with a fixed point). 


