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Conjugation spaces

Jean-Claude Hausmann, Tara S Holm and Volker Puppe

Algebraic & Geometric Topology 5 (2005) 923–964

arXiv: math.AT/0412057


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.

cohomology rings, equivariant cohomology, spaces with involution, real spaces
Mathematical Subject Classification 2000
Primary: 55N91, 55M35
Secondary: 53D05, 57R22
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Received: 16 February 2005
Accepted: 7 July 2005
Published: 5 August 2005
Jean-Claude Hausmann
Section de mathématiques
2–4 rue du Lièvre
CP 64 CH-1211 Genève 4
Tara S Holm
Department of Mathematics
University of Connecticut
Storrs CT 06269-3009
Volker Puppe
Universität Konstanz
Fakultät für Mathematik
Fach D202
D-78457 Konstanz