# Pontryagin numbers on manifolds with an $S^1$-action

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

• What's the formula for $4$-manifolds with $S^1$ action? Dec 12 '13 at 0:41
• The formula can be deduced from $G$-signature theorem, you can check here: maths.ed.ac.uk/~aar/papers/hirzrem.pdf page 13 Dec 12 '13 at 0:52