scholarly journals Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points

2014 ◽  
Vol 21 (4) ◽  
pp. 691-696 ◽  
Author(s):  
Yunhyung Cho ◽  
Min Kyu Kim
Keyword(s):  
Fixed Points ◽  
Betti Numbers ◽  
Circle Action

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].

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].

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.

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 Face Rings of Cycles, Associahedra, and Standard Young Tableaux

10.37236/5208 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Anton Dochtermann
Keyword(s):  
Fixed Points ◽  
Morse Theory ◽  
Betti Numbers ◽  
Free Resolution ◽  
Discrete Morse Theory ◽  
Young Tableaux ◽  
N Cycle ◽  
3 Dimensional ◽  
Cellular Resolution ◽  
Standard Young Tableaux

We show that $J_n$, the Stanley-Reisner ideal of the $n$-cycle, has a free resolution supported on the $(n-3)$-dimensional simplicial associahedron $A_n$. This resolution is not minimal for $n \geq 6$; in this case the Betti numbers of $J_n$ are strictly smaller than the $f$-vector of $A_n$. We show that in fact the Betti numbers $\beta_{d}$ of $J_n$ are in bijection with the number of standard Young tableaux of shape $(d+1, 2, 1^{n-d-3})$. This complements the fact that the number of $(d-1)$-dimensional faces of $A_n$ are given by the number of standard Young tableaux of (super)shape $(d+1, d+1, 1^{n-d-3})$; a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of $J_n$ that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.

On $\alpha$-nonexpansive mapping and fixed-points in Banach spaces

2018 ◽  
Vol 2018 (-) ◽  
Author(s):  
Prondanai Kaskasem ◽  
Chakkrid Klin-eam ◽  
Suthep Suantai
Keyword(s):  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Banach Spaces

Fixed Points of Sequences of Mappings

2018 ◽  
Vol 2018 (-) ◽  
Author(s):  
C. Ganesa Moorthy ◽  
S. Iruthaya Raj
Keyword(s):  
Fixed Points
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