FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

2000 ◽  
Vol 02 (01) ◽  
pp. 75-86 ◽  
Author(s):  
FUQUAN FANG ◽  
XIAOCHUN RONG
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Fundamental Groups ◽  
Vanishing Theorem ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Finite Fundamental Group ◽  
Finiteness Theorems ◽  
Point Set ◽  
Simply Connected

We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any Tk-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ; ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.

On the fundamental group of an orbit space

1965 ◽  
Vol 61 (3) ◽  
pp. 639-646 ◽  
Author(s):  
M. A. Armstrong
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Fundamental Group ◽  
Simplicial Complex ◽  
Orbit Space ◽  
Fixed Point Set ◽  
Point Set ◽  
Simply Connected

Introduction. Let K be a connected simplicial complex, finite or infinite, its polyhedron ((2), page 45) being the space X. Then X is connected. Suppose further that X is simply connected. For any group G of simplicial transformations of X, H will denote the normal subgroup generated by elements which have a non-empty fixed-point set.

Rationale for a Localization Formula

2020 ◽  
pp. 232-238
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Closed Form ◽  
Blow Up ◽  
Torus Action ◽  
Exact Form ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Localization Formula ◽  
Point Set ◽  
Finite Sum

This chapter offers a rationale for a localization formula. It looks at the equivariant localization formula of Atiyah–Bott and Berline–Vergne. The equivariant localization formula of Atiyah–Bott and Berline–Vergne expresses, for a torus action, the integral of an equivariantly closed form over a compact oriented manifold as a finite sum over the fixed point set. The central idea is to express a closed form as an exact form away from finitely many points. Throughout his career, Raoul Bott exploited this idea to prove many different localization formulas. The chapter then considers circle actions with finitely many fixed points. It also studies the spherical blow-up.

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.

scholarly journals Circle actions on homotopy spheres with codimension 4 fixed point set

1983 ◽  
Vol 109 (2) ◽  
pp. 349-362 ◽  
Author(s):  
Ronald Fintushel ◽  
Peter Pao
Keyword(s):  
Fixed Point ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Point Set

A fixed point index for maps with Borsuk-presentable fixed point set

1982 ◽  
Vol 38 (1) ◽  
pp. 549-555
Author(s):  
Christian C. Fenske
Keyword(s):  
Fixed Point ◽  
Fixed Point Index ◽  
Point Index ◽  
Fixed Point Set ◽  
Point Set

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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