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2005 Conjugation spaces
Jean-Claude Hausmann, Tara S Holm, Volker Puppe
Algebr. Geom. Topol. 5(3): 923-964 (2005). DOI: 10.2140/agt.2005.5.923

Abstract

There are classical examples of spaces X with an involution τ whose mod 2 cohomology ring resembles that of their fixed point set Xτ: there is a ring isomorphism κ:H2(X)H(Xτ). 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 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 XT 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 conjugate-equivariant 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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Jean-Claude Hausmann. Tara S Holm. Volker Puppe. "Conjugation spaces." Algebr. Geom. Topol. 5 (3) 923 - 964, 2005. https://doi.org/10.2140/agt.2005.5.923

Information

Received: 16 February 2005; Accepted: 7 July 2005; Published: 2005
First available in Project Euclid: 20 December 2017

zbMATH: 1081.55006
MathSciNet: MR2171799
Digital Object Identifier: 10.2140/agt.2005.5.923

Subjects:
Primary: 55M35, 55N91
Secondary: 53D05, 57R22

Rights: Copyright © 2005 Mathematical Sciences Publishers

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