On an algebraic formula and applications to group action on manifolds

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 $p$ is a prime.Comment: 7 pages, revised slightly to update a new reference and reassign the credit of the idea in this not

Some remarks on circle action on manifolds

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.Comment: 10 pages,to appear in Mathematical Research Letter

Circle action and some vanishing results on manifolds

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 Witten-Taubes-Bott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of Conner-Floyd, Landweber-Stong and Hirzebruch-Slodowy.Comment: 7 pages, typos corrected and minors modifie

Boltje-Maisch resolutions of Specht modules

In \cite{21}, Boltje and Maisch found a permutation complex of Specht modules in representation theory of Hecke algebras, which is the same as the Boltje-Hartmann complex appeared in the representation theory of symmetric groups and general linear groups. In this paper we prove the exactness of Boltje-Maisch complex in the dominant weight case.Comment: 17 page