Some remarks on circle action on manifolds

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 $\lambda =({\lambda }_1,...,{\lambda }_u)$ of weight $m$ such that the Chern number $(c_{{\lambda }_1}...c_{{\lambda }_u}{)}^n[M]$ (resp. Pontrjagin number $(p_{{\lambda }_1}...p_{{\lambda }_u}{)}^n[N]$) 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.

10 pages,to appear in Mathematical Research Letters

Cited by in corpus (2)

• Circle action, lower bound of fixed points and characteristic numbers
• On circle actions with exactly three fixed points