In this paper we consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of \mathbb {Z} _p action on manifolds with isolated fixed points when \mathbb {Z} _p is a prime.

Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable WittenTaubesBott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of ConnerFloyd, LandweberStong and HirzebruchSlodowy.

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 M^{2mn} ), if there exists a partition M^{2mn} of weight M^{2mn} such that the Chern number M^{2mn} (resp. Pontrjagin number M^{2mn} ) is nonzero, then\emph {any} circle action on M^{2mn} (resp. M^{2mn} ) has at least M^{2mn} fixed points. When an even-dimensional smooth manifold M^{2mn} admits a semi-free action with isolated fixed points, we show that M^{2mn} 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.

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck’s Torus Theorem, and obtain corresponding results for p-compact groups and p-Noetherian groups.