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Conjugation spaces 

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Published in  Algebraic & Geometric Topology. 2005, vol. 5, p. 923964  
Abstract  There are classical examples of spaces X with an involution tau whose mod 2comhomology ring resembles that of their fixed point set X^tau: there is a ring isomorphism kappa: H^2*(X) > H^*(X^tau). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism kappa 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 antisymplectic 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 coadjoint orbits of any semisimple 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 kappa maps the Chern classes onto the StiefelWhitney classes of the fixed bundle.  
Identifiers  arXiv: math/0412057v2  
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Citation (ISO format)  HAUSMANN, JeanClaude, HOLM, Tara, PUPPE, Volker. Conjugation spaces. In: Algebraic & Geometric Topology, 2005, vol. 5, p. 923964. https://archiveouverte.unige.ch/unige:12195 