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 Betti numbers of fixed point sets and multiplicities of indecomposable summands

2003 ◽  
Vol 74 (2) ◽  
pp. 165-172
Author(s):  
Semra Öztürk Kaptanoglu
Keyword(s):  
Fixed Point ◽  
Equivariant Cohomology ◽  
Betti Numbers ◽  
Localization Theorem ◽  
Fixed Point Set ◽  
Moore Space ◽  
Even Order ◽  
Point Set ◽  
Characteristic 2 ◽  
A Finite Group

AbstractLet G be a finite group of even order, k be a field of characteristic 2, and M be a finitely generated kG-module. If M is realized by a compact G-Moore space X, then the Betti numbers of the fixed point set XCn and the multiplicities of indecomposable summands of M considered as a kCn-module are related via a localization theorem in equivariant cohomology, where Cn is a cyclic subgroup of G of order n. Explicit formulas are given for n = 2 and n = 4.

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.

Cohomology Ring of Symplectic Quotients by Circle Actions

2004 ◽  
Vol 56 (3) ◽  
pp. 553-565 ◽  
Author(s):  
Ramin Mohammadalikhani
Keyword(s):  
Fixed Point ◽  
Cohomology Ring ◽  
Circle Action ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Generating Set ◽  
Special Cases ◽  
Symplectic Quotients ◽  
Point Set

AbstractIn this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be “formally Hamiltonian”. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.

The central sphere of an ALE space

2020 ◽  
Author(s):  
Nigel Hitchin
Keyword(s):  
Fixed Point ◽  
Algebraic Geometry ◽  
Rational Curves ◽  
Algebraic Surfaces ◽  
Circle Action ◽  
Fixed Point Set ◽  
Point Set ◽  
Ale Space ◽  
Central Sphere

Abstract We consider the induced metric on the spherical fixed point set of a circle action on an ALE space and describe it by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

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