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

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

One can find the Pontryagin numbers. It is an application of a more general version of the G-signature theorem, when one considers the signature operator on M twisted by a vector bundle. It is Theorem 8.11 respectively formula 8.12 in the paper which proves the G-signature theorem:

Atiyah, M. F.; Singer, I. M. The index of elliptic operators. III. Ann. of Math. (2) 87 1968 546–604.

(Actually you provided a link to an electronic version in your question.)


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