Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all Pontryagin numbers of the manifold? I would be grateful for some nice reference on this topic.

For four-manifolds such a formula for $p_1$ exists but don't know if it exists for higher dimensions. Indeed, $p_1=3\tau$, where $\tau$ is the signature, and the signature can be calculated by proposition 6.18 here http://www.ma.utexas.edu/users/dafr/Index/asindiii.pdf