Topological obstructions to certain Lie group actions on manifolds

Given a smooth closed $S^{1}$-manifold $M$, this article studies the extent to which certain numbers of the form $\left( f^{\ast}\left( x\right) \cdot P\cdot C\right) \left[ M\right]$ are determined by the fixed-point set $M^{S^{1}}$, where $f:M\rightarrow K\left( \pi_{1}\left( M\right), 1\right)$ classifies the universal cover of $M$, $x\in H^{\ast}\left( \pi_{1}\left( M\right) ;\mathbb{Q}\right)$, $P$ is a polynomial in the Pontrjagin classes of $M$, and $C$ is in the subalgebra of $H^{\ast}\left( M;\mathbb{Q}\right)$ generated by $H^{2}\left( M;\mathbb{Q}\right)$. When $M^{S^{1}}=\varnothing$, various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free $S^{1}$-action extends to an action by some semisimple compact Lie group $G$, then $\left( f^{\ast}(x)\cdot P\cdot C\right) [M]=0$. Similar vanishing results are obtained for spin manifolds admitting certain $S^{1}$-actions.