CIRCLE ACTION AND SOME VANISHING RESULTS ON MANIFOLDS

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.

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Cohomology Ring of Symplectic Quotients by Circle Actions

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-025-9 ◽

2004 ◽

Vol 56 (3) ◽

pp. 553-565 ◽

Cited By ~ 1

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.

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Rationale for a Localization Formula

Introductory Lectures on Equivariant Cohomology ◽

10.23943/princeton/9780691191751.003.0029 ◽

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.

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FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

Communications in Contemporary Mathematics ◽

10.1142/s0219199700000062 ◽

2000 ◽

Vol 02 (01) ◽

pp. 75-86 ◽

Cited By ~ 4

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.

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Hamiltonian circle actions with fixed point set almost minimal

Mathematische Zeitschrift ◽

10.1007/s00209-019-02236-6 ◽

2019 ◽

Vol 293 (3-4) ◽

pp. 1315-1336

Author(s):

Hui Li

Keyword(s):

Fixed Point ◽

Fixed Point Set ◽

Circle Actions ◽

Point Set

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Borel Localization for a Circle Action

Introductory Lectures on Equivariant Cohomology ◽

10.23943/princeton/9780691191751.003.0026 ◽

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.

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The central sphere of an ALE space

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haaa051 ◽

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.

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Limiting cases of Boardman's five halves theorem

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091513000400 ◽

2013 ◽

Vol 56 (3) ◽

pp. 723-732 ◽

Cited By ~ 1

Author(s):

Michael C. Crabb ◽

Pedro L. Q. Pergher

Keyword(s):

Fixed Point ◽

Dimensional Manifold ◽

Fixed Point Set ◽

Minimal Dimension ◽

Point Set ◽

Image Position

AbstractThe famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and ifis the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).

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