Let $X$ be a smooth manifold with an action of a Lie group $G$, i.e. at least we have a map $G \times X \to X$ of smooth manifolds. Then apply the cohomology functor and obtain $H^i (X, \mathbb{R}) \to H^i (G \times X, \mathbb{R}).$ Is it possible to compute this map?
I am mainly interested in the case of $G=S^1$, so this map looks like $H^i (X, \mathbb{R}) \to H^i (S^1 \times X, \mathbb{R}) = H^{i} (X, \mathbb{R}) \oplus H^{i-1} (X, \mathbb{R}),$ but what is this map? For me it is enough to understand this map for $X=T^* Y$, i.e. when $X$ is a cotangent bundle of a smooth manifold with a natural action of the circle on its fibers.
I’ll be grateful for any references.