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.

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.

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.

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.

scholarly journals On the localization formula in equivariant cohomology

2007 ◽  
Vol 154 (7) ◽  
pp. 1493-1501 ◽  
Author(s):  
Andrés Pedroza ◽  
Loring W. Tu
Keyword(s):  
Equivariant Cohomology ◽  
Localization Formula

Localization Formulas

2020 ◽  
pp. 238-244
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Euler Class ◽  
The Other ◽  
Torus Action ◽  
Characteristic Classes ◽  
Fixed Point Set ◽  
Localization Formula ◽  
Point Set ◽  
The One ◽  
Finite Sum

This chapter highlights localization formulas. The equivariant localization formula for a torus action expresses the integral of an equivariantly closed form as a finite sum over the fixed point set. It was discovered independently by Atiyah and Bott on the one hand, and Berline and Vergne on the other, around 1982. The chapter describes the equivariant localization formula for a circle action and works out an application to the surface area of a sphere. It also explores some equivariant characteristic classes of a vector bundle. These include the equivariant Euler class, the equivariant Pontrjagin classes, and the equivariant Chern classes.

A Crash Course in Representation Theory

2020 ◽  
pp. 222-228
Author(s):  
Loring W. Tu
Keyword(s):  
Group Theory ◽  
Representation Theory ◽  
Equivariant Cohomology ◽  
Matrix Multiplication ◽  
Invariant Subspaces ◽  
Group Homomorphism ◽  
Minimal Representation ◽  
Real Representation ◽  
Localization Formula

This chapter studies representation theory. In order to state the equivariant localization formula of Atiyah–Bott and Berline–Vergne, one will need to know some representation theory. Representation theory “represents” the elements of a group by matrices in such a way that group multiplication becomes matrix multiplication. It is a way of simplifying group theory. The chapter provides the minimal representation theory needed for equivariant cohomology. A real representation of a group G is a group homomorphism. Every representation has at least two invariant subspaces, 0 and V. These are called the trivial invariant subspaces. A representation is said to be irreducible if it has no invariant subspaces other than 0 and V; otherwise, it is reducible.

scholarly journals LIFTS OF SMOOTH GROUP ACTIONS TO LINE BUNDLES

2001 ◽  
Vol 33 (3) ◽  
pp. 351-361 ◽  
Author(s):  
IGNASI MUNDET I RIERA
Keyword(s):  
Invariant Theory ◽  
Compact Manifold ◽  
Symplectic Geometry ◽  
Equivariant Cohomology ◽  
Line Bundles ◽  
Complex Line ◽  
Geometric Invariant ◽  
Smooth Action ◽  
Smooth Group Actions ◽  
First Chern Class

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H2G(X; ℤ), and that the different lifts of the action are classified by the lifts of c1(L) to H2G(X; ℤ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L, and which has a G-invariant curvature, then there is a lift of the action to a certain power Ld (where d is independent of L) which leaves fixed the induced metric on Ld and the connection ∇[otimes ]d. This generalises to symplectic geometry a well-known result in geometric invariant theory.

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