Topological obstructions to certain Lie group actions on manifolds

Abstract:

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.