Mathematical Research Letters

Volume 18 (2011)

Number 3

Some Remarks on Circle Action on Manifolds

Pages: 437 – 446

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n3.a5

Authors

Ping Li (Tongji University, Shanghai 200092, China)

Kefeng Liu (University of California at Los Angeles, Los Angeles, CA 90095, USA and Center of Mathematical Science, Zhejiang University, 310027, China)

Abstract

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 $N^{4mn}$), if there exists a partition $\lambda=(\lambda_{1},\cdots,\lambda_{u})$ of weight $m$ such that the Chern number $(c_{\lambda_{1}}\cdots c_{\lambda_{u}})^{n}[M]$ (resp. Pontrjagin number $(p_{\lambda_{1}}\cdots p_{\lambda_{u}})^{n}[N]$) is nonzero, then \emph{any} circle action on $M^{2mn}$ (resp. $N^{4mn}$) has at least $n+1$ fixed points. When an even-dimensional smooth manifold $N^{2n}$ admits a semi-free action with isolated fixed points, we show that $N^{2n}$ 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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