科研之友
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摘要
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 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 nonempty isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Bott's residue formula and rigidity theorem.
