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.

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.

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.

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.

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.

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
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