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# 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

#### 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.