Overview

2020 ◽  
pp. 5-10
Author(s):  
Loring W. Tu
Keyword(s):  
Twentieth Century ◽  
Fixed Points ◽  
Group Action ◽  
Equivariant Cohomology ◽  
Topological Spaces ◽  
Computational Tool ◽  
Geometric Problem ◽  
Localization Formula ◽  
Algebraic Problem ◽  
Finite Sum

This chapter provides an overview of equivariant cohomology. Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces to rings, cohomology turns a geometric problem into an easier algebraic problem. Equivariant cohomology is a cohomology theory that takes into account the symmetries of a space. Many topological and geometrical quantities can be expressed as integrals on a manifold. Integrals are vitally important in mathematics. However, they are also rather difficult to compute. When a manifold has symmetries, as expressed by a group action, in many cases the localization formula in equivariant cohomology computes the integral as a finite sum over the fixed points of the action, providing a powerful computational tool.

Some Applications

2020 ◽  
pp. 252-258
Author(s):  
Loring W. Tu
Keyword(s):  
Symplectic Geometry ◽  
Equivariant Cohomology ◽  
Invariant Form ◽  
Computational Tool ◽  
Localization Formula ◽  
Finite Sum

This chapter explores some applications of equivariant cohomology. Since its introduction in the Fifties, equivariant cohomology has found applications in topology, symplectic geometry, K-theory, and physics, among other fields. Its greatest utility may be in converting an integral on a manifold to a finite sum. Since many problems in mathematics can be expressed in terms of integrals, the equivariant localization formula provides a powerful computational tool. The chapter then discusses a few of the applications of the equivariant localization formula. In order to use the equivariant localization formula to compute the integral of an invariant form, the form must have an equivariantly closed extension.

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.

Introductory Lectures on Equivariant Cohomology

2020 ◽  
Author(s):  
Loring W. Tu
Keyword(s):  
Group Action ◽  
Algebraic Topology ◽  
Equivariant Cohomology ◽  
Differential Forms ◽  
Localization Theorem ◽  
Fixed Point Set ◽  
De Rham ◽  
Topological Construction ◽  
Finite Sum ◽  
Manifold Theory

Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah–Bott and Berline–Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, the book begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.

Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields

2015 ◽  
Vol 25 (11) ◽  
pp. 1550150 ◽  
Author(s):  
Oxana Cerba Diaconescu ◽  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Parameter Space ◽  
Group Action ◽  
Sufficient Conditions ◽  
Phase Portraits ◽  
Topological Spaces ◽  
Canonical Forms ◽  
Affine Transformations ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Quotient Spaces

In this article, we consider the class [Formula: see text] of all real quadratic differential systems [Formula: see text], [Formula: see text] with gcd (p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class [Formula: see text] so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

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 Coupled fixed points of weakly F-contractive mappings in topological spaces

2011 ◽  
Vol 24 (7) ◽  
pp. 1185-1190 ◽  
Author(s):  
Yeol Je Cho ◽  
Masood Hussain Shah ◽  
Nawab Hussain
Keyword(s):  
Fixed Points ◽  
Topological Spaces ◽  
Contractive Mappings ◽  
Coupled Fixed Points

scholarly journals Fixed points for pairs of mappings indcomplete topological spaces

1993 ◽  
Vol 16 (2) ◽  
pp. 259-266 ◽  
Author(s):  
Troy L. Hicks ◽  
B. E. Rhoades
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Large Class ◽  
Distance Function ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Triangular Inequality

Several important metric space fixed point theorems are proved for a large class of non-metric spaces. In some cases the metric space proofs need only minor changes. This is surprising since the distance function used need not be symmetric and need not satisfy the triangular inequality.

scholarly journals Realizing doubles: a conjugation zoo

2020 ◽  
pp. 1-16
Author(s):  
Wolfgang Pitsch ◽  
Jérôme Scherer
Keyword(s):  
Vector Space ◽  
Fixed Points ◽  
Projective Spaces ◽  
Topological Spaces ◽  
Euclidean Spaces ◽  
Complex Conjugation ◽  
Complex Projective Spaces ◽  
Complex Projective

Abstract Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod 2 cohomology (as a graded vector space, a ring and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct ‘exotic’ conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such.

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