Equivariant Cohomology of S1-Actions on 4-Manifolds

2007 ◽  
Vol 50 (3) ◽  
pp. 365-376 ◽  
Author(s):  
Leonor Godinho
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Equivariant Cohomology ◽  
Circle Action ◽  
Dimensional Manifold ◽  
Point Data

AbstractLet M be a symplectic 4-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

scholarly journals On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

2017 ◽  
Vol 19 (04) ◽  
pp. 1750043 ◽  
Author(s):  
Silvia Sabatini
Keyword(s):  
Fixed Points ◽  
Complex Manifold ◽  
Index Formula ◽  
Circle Action ◽  
Hilbert Polynomial ◽  
Dimensional Manifold ◽  
Hamiltonian Action ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Chern Numbers

Let [Formula: see text] be a compact, connected, almost complex manifold of dimension [Formula: see text] endowed with a [Formula: see text]-preserving circle action with isolated fixed points. In this paper, we analyze the “geography problem” for such manifolds, deriving equations relating the Chern numbers to the index [Formula: see text] of [Formula: see text]. We study the symmetries and zeros of the Hilbert polynomial, which imply many rigidity results for the Chern numbers when [Formula: see text]. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a 6-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. Another consequence is that, if the action is Hamiltonian, the minimal Chern number coincides with the index and is at most [Formula: see text].

scholarly journals CIRCLE ACTION AND SOME VANISHING RESULTS ON MANIFOLDS

2011 ◽  
Vol 22 (11) ◽  
pp. 1603-1610 ◽  
Author(s):  
PING LI ◽  
KEFENG LIU
Keyword(s):  
Fixed Point ◽  
Rigidity Theorem ◽  
Circle Action ◽  
Dimensional Manifold ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Signature Operator ◽  
Spin Manifolds ◽  
Point Set

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.

Proof of the Localization Formula for a Circle Action

2020 ◽  
pp. 244-252
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Points ◽  
Closed Form ◽  
Equivariant Cohomology ◽  
Blow Up ◽  
Free Action ◽  
Circle Action ◽  
Exact Form ◽  
Localization Formula ◽  
Blowing Up

This chapter provides a proof of the localization formula for a circle action. It evaluates the integral of an equivariantly closed form for a circle action by blowing up the fixed points. On the spherical blow-up, the induced action has no fixed points and is therefore locally free. The spherical blow-up is a manifold with a union of disjoint spheres as its boundary. For a locally free action, one can express an equivariantly closed form as an exact form. Since the localized equivariant cohomology of a locally free action is zero, after localization an equivariantly closed form must be equivariantly exact. Stokes's theorem then reduces the integral to a computation over spheres.

Borel Localization for a Circle Action

2020 ◽  
pp. 213-220
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Equivariant Cohomology ◽  
Ring Structure ◽  
Circle Action ◽  
Localization Theorem ◽  
Fixed Point Set ◽  
Ring Isomorphism ◽  
Point Set

This chapter explores Borel localization for a circle action. For a circle action, the Borel localization theorem says that up to torsion, the equivariant cohomology of an S1-manifold is concentrated on its fixed point set and that the isomorphism in localized equivariant cohomology of the manifold and its fixed point set is a ring isomorphism. This is clearly an important result in its own right. Moreover, since the fixed point set is a regular submanifold and is usually simpler than the manifold, the Borel localization theorem sometimes allows one to obtain the ring structure of the equivariant cohomology of an S1-manifold from that of its fixed point set. The chapter demonstrates this method with the example of S1 acting on S2 by rotations.

scholarly journals Unimodality of Betti numbers for Hamiltonian circle actions with index-increasing moment Maps

2016 ◽  
Vol 27 (05) ◽  
pp. 1650043 ◽  
Author(s):  
Yunhyung Cho
Keyword(s):  
Fixed Points ◽  
Morse Index ◽  
Equivariant Cohomology ◽  
Arbitrary Dimension ◽  
Betti Numbers ◽  
Circle Action ◽  
Dimensional Case ◽  
Circle Actions ◽  
Moment Maps ◽  
The Moment

The unimodality conjecture posed by Tolman in [L. Jeffrey, T. Holm, Y. Karshon, E. Lerman and E. Meinrenken, Moment maps in various geometries, http://www.birs.ca/workshops/2005/05w5072/report05w5072.pdf ] states that if [Formula: see text] is a [Formula: see text]-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers [Formula: see text] is unimodal, i.e. [Formula: see text] for every [Formula: see text]. Recently, the author and Kim [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] proved that the unimodality holds in eight-dimensional case by using equivariant cohomology theory. In this paper, we generalize the idea in [Y. Cho and M. Kim, Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points, Math. Res. Lett. 21(4) (2014) 691–696] to an arbitrary dimensional case. We prove the conjecture in arbitrary dimension under the assumption that the moment map [Formula: see text] is index-increasing, which means that [Formula: see text] implies [Formula: see text] for every pair of critical points [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the Morse index of [Formula: see text] with respect to [Formula: see text].

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

Coincidence and fixed points for hybrid tangential maps

2010 ◽  
Vol 17 (2) ◽  
pp. 273-285
Author(s):  
Tayyab Kamran ◽  
Quanita Kiran
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Property ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contractive Condition ◽  
The Common

Abstract In [Int. J. Math. Math. Sci. 2005: 3045–3055] by Liu et al. the common property (E.A) for two pairs of hybrid maps is defined. Recently, O'Regan and Shahzad [Acta Math. Sin. (Engl. Ser.) 23: 1601–1610, 2007] have introduced a very general contractive condition and obtained some fixed point results for hybrid maps. We introduce a new property for pairs of hybrid maps that contains the property (E.A) and obtain some coincidence and fixed point theorems that extend/generalize some results from the above-mentioned papers.

scholarly journals Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 501
Author(s):  
Ahmed Boudaoui ◽  
Khadidja Mebarki ◽  
Wasfi Shatanawi ◽  
Kamaleldin Abodayeh
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
System Of Differential Equations ◽  
Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

scholarly journals CONFORMAL HOUSE

2010 ◽  
Vol 25 (24) ◽  
pp. 4603-4621 ◽  
Author(s):  
THOMAS A. RYTTOV ◽  
FRANCESCO SANNINO
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Gauge Group ◽  
Gauge Theories ◽  
Simultaneous Presence ◽  
Consistency Check ◽  
Effective Lagrangian ◽  
Anomalous Dimensions ◽  
Mass Terms

We investigate the gauge dynamics of nonsupersymmetric SU (N) gauge theories featuring the simultaneous presence of fermionic matter transforming according to two distinct representations of the underlying gauge group. We bound the regions of flavors and colors which can yield a physical infrared fixed point. As a consistency check we recover the previously investigated bounds of the conformal windows when restricting to a single matter representation. The earlier conformal windows can be imagined to be part now of the new conformal house. We predict the nonperturbative anomalous dimensions at the infrared fixed points. We further investigate the effects of adding mass terms to the condensates on the conformal house chiral dynamics and construct the simplest instanton induced effective Lagrangian terms.

scholarly journals Common fixed points of single-valued and multivalued maps

2005 ◽  
Vol 2005 (19) ◽  
pp. 3045-3055 ◽  
Author(s):  
Yicheng Liu ◽  
Jun Wu ◽  
Zhixiang Li
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Common Fixed Points ◽  
Partial Answer ◽  
Multivalued Maps ◽  
Hybrid Pair ◽  
Common Fixed Point Theorems

We define a new property which contains the property (EA) for a hybrid pair of single- and multivalued maps and give some new common fixed point theorems under hybrid contractive conditions. Our results extend previous ones. As an application, we give a partial answer to the problem raised by Singh and Mishra.

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