Conjugation spaces

Hausmann, Jean-Claude¹ ; Holm, Tara S² ; Puppe, Volker³

¹Section de mathématiques, 2–4 rue du Lièvre, CP 64 CH-1211 Genève 4, Switzerland
²Department of Mathematics, University of Connecticut, Storrs CT 06269-3009, USA
³Universität Konstanz, Fakultät für Mathematik, Fach D202, D-78457 Konstanz, Germany

Algebraic and Geometric Topology, Tome 5 (2005) no. 3, pp. 923-964

Résumé

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 ℂk 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.

Détail

DOI : 10.2140/agt.2005.5.923

Keywords: cohomology rings, equivariant cohomology, spaces with involution, real spaces

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