Some remarks on circle action on manifolds

Ping Li Kefeng Liu

Algebraic Topology and General Topology mathscidoc:1912.43070

arXiv preprint arXiv:1008.4826, 2010.8
This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold M^{2mn} (resp. a smooth manifold M^{2mn} ), if there exists a partition M^{2mn} of weight M^{2mn} such that the Chern number M^{2mn} (resp. Pontrjagin number M^{2mn} ) is nonzero, then\emph {any} circle action on M^{2mn} (resp. M^{2mn} ) has at least M^{2mn} fixed points. When an even-dimensional smooth manifold M^{2mn} admits a semi-free action with isolated fixed points, we show that M^{2mn} bounds, which generalizes a well-known fact in the free case. We also provide a topological obstruction, in terms of the first Chern class, to the existence of semi-free circle action with\emph {nonempty} isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Bott's residue formula and rigidity theorem.
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  title={Some remarks on circle action on manifolds},
  author={Ping Li, and Kefeng Liu},
  booktitle={arXiv preprint arXiv:1008.4826},
Ping Li, and Kefeng Liu. Some remarks on circle action on manifolds. 2010. In arXiv preprint arXiv:1008.4826.
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