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.
We prove a homological stability theorem for unlinked circles in $3$-manifolds and give an application to certain groups of diffeomorphisms of 3-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 $λ=(λ_{1},...,λ_{u})$ of weight $m$ such that the Chern number $(c_{λ_{1}}... c_{λ_{u}})^{n}[M]$ (resp. Pontrjagin number $(p_{λ_{1}}... p_{λ_{u}})^{n}[N]$) 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 point…
We give a lower bound for the degree of a finite cover of a hyperbolic 3-manifold which fibers over the circle, in terms of volume, the diameter of the manifold and other new invariants.
We establish an S^1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S^1-actions on closed spin manifolds to the case of Z/k-manifolds.
We determine the homeomorphism (resp. diffeomorphism) types of those 2-connected 7-manifolds (resp. smooth 2-connected 7-manifolds) that admit regular circle actions (resp. smooth regular circle actions).
In the paper we describe obstructions for the existence of symplectic and Hamiltonian symplectic circle actions on closed compact manifolds in terms of Hirzebruch genera and relations between differential and homotopic invariants of such manifolds.
The classification of Seifert manifolds was given in terms of numeric data by Seifert in 1933, and then generalized by Orlik and Raymond in 1968 to circle actions on closed 3d manifolds. In this paper, we further generalize the classification to circle actions on 3d manifolds with boundaries by adding a numeric parameter and a union of cycle graphs. Then we describe the equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincaré series for these manifolds and discuss the equivariant form…
Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topolog…
A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an n-cube with a semifree circle action and isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This …
Assume $(M, ω)$ is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case $\dim$ $H^2(M)<3$. We give a complete list of the possible manifolds, determine their equivariant cohomology ring and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.
In this paper, we first establish a K-theory version of the equivariant family index theorem for a circle action, then use it to prove several rigidity and vanishing theorems on the equivariant K-theory level.
We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits.
Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.
We show that almost complex circle actions with exactly three fixed points do not exist in dimension 8 and present an infinite series of 6-dimensional manifolds possessing an almost complex circle action with exactly two fixed points.
The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. Using this correspondence and combinatorial arguments, we prove that a simply connected 6-dimensional torus manifold with vanishing odd degree cohomology is equivariantly diffeomorphic to the 6-dimensional sphere or an equivariant connected sum of copies of 6-dimensional quasitor…
