The Cohomology Rings of Symplectic Quotients

Tolman, Susan; Weitsman, Jonathan

doi:10.4310/cag.2003.v11.n4.a6

Cited by 58 publications

(71 citation statements)

References 15 publications

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“…Indeed, we believe this K-theoretic extension to be more natural than a corresponding extension to integral cohomology. As we mentioned above, when working with integral cohomology, one needs to place additional constraints on the torsion to establish the Atiyah-Bott lemma and its consequences (see [39]), but such torsion constraints are not present in the K-theoretic version.…”

Section: Introductionmentioning

confidence: 99%

Surjectivity for Hamiltonian 𝐺-spaces in 𝐾-theory

Harada

Landweber

2007

Trans. Amer. Math. Soc.

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Abstract. Let G be a compact connected Lie group, and (M, ω) a Hamiltonian G-space with proper moment map µ. We give a surjectivity result which expresses the K-theory of the symplectic quotient M/ /G in terms of the equivariant K-theory of the original manifold M , under certain technical conditions on µ. This result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K-theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G-spaces. We discuss this lemma in detail and highlight the differences between the K-theory and rational cohomology versions of this lemma.We also introduce a K-theoretic version of equivariant formality and prove that when the fundamental group of G is torsion-free, every compact Hamiltonian G-space is equivariantly formal. Under these conditions, the forgetful mapis surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in H 2

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Section: Introductionmentioning

confidence: 99%

Surjectivity for Hamiltonian 𝐺-spaces in 𝐾-theory

Harada

Landweber

2007

Trans. Amer. Math. Soc.

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“…Under additional assumptions on the torsion of the fixed point sets and the group action, this map is surjective over the integers or Z 2 as well. There are several ways to compute the kernel of K. Tolman and Weitsman [27] did so in the way that is most suited to our needs.…”

Section: Symplectic Reductionsmentioning

confidence: 99%

Conjugation spaces

Hausmann¹,

Holm

Puppe

2005

Algebr. Geom. Topol.

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There are classical examples of spaces X with an involution τ whose mod2-comhomology ring resembles that of their fixed point set X τ : there is a ring isomorphism κ :. Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism κ is part of an interesting structure in equivariant cohomology called an H * -frame. An H * -frame, if it exists, is natural and unique. A space with involution admitting an H * -frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C k with the complex conjugation. A compact symplectic manifold, with an anti-symplectic involution compatible with a Hamiltonian action of a torus T , is a conjugation space, provided X T is itself a conjugation space. This includes the co-adjoint orbits of any semi-simple compact Lie group, equipped with the Chevalley involution. We also study conjugateequivariant complex vector bundles ("real bundles" in the sense of Atiyah) over a conjugation space and show that the isomorphism κ maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.

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“…In [19] the authors note that under reasonable assumptions about the torsion of the fixed point sets and the group action, this map is surjective over the integers as well. For T = S 1 one can state the result as follows.…”

Section: Introductionmentioning

confidence: 99%

Real loci of symplectic reductions

Goldin

Holm

2004

Trans. Amer. Math. Soc.

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Abstract. Let M be a compact, connected symplectic manifold with a Hamiltonian action of a compact n-dimensional torus T . Suppose that M is equipped with an anti-symplectic involution σ compatible with the T -action. The real locus of M is the fixed point set M σ of σ. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction M//T has the same ordinary cohomology as its real locus (M//T ) σ red , with degrees halved. This extends Duistermaat's original result on real loci to a case in which there is not a natural Hamiltonian torus action.

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The Cohomology Rings of Symplectic Quotients

Cited by 58 publications

References 15 publications

Surjectivity for Hamiltonian 𝐺-spaces in 𝐾-theory

Surjectivity for Hamiltonian 𝐺-spaces in 𝐾-theory

Conjugation spaces

Real loci of symplectic reductions

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