Some remarks on circle action on manifolds

TL;DR

This paper investigates circle actions on almost-complex and smooth manifolds, establishing conditions for fixed points, bounding properties, and obstructions using topological invariants and advanced formulas.

Contribution

It introduces new criteria linking topological invariants to fixed points of circle actions and extends known results on manifold boundaries and obstructions.

Findings

Nonzero Chern or Pontrjagin numbers imply at least n+1 fixed points for circle actions.

Manifolds with semi-free actions and isolated fixed points bound, generalizing known free action results.

Topological obstructions related to the first Chern class prevent certain semi-free circle actions.

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.