Algebraic and Geometric Topology 5 (2005), paper no. 39, pages 923-964.

Conjugation spaces

Jean-Claude Hausmann, Tara Holm and Volker Puppe

Abstract. There are classical examples of spaces X with an involution tau whose mod 2-comhomology 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 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 conjugate-equivariant 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 Stiefel-Whitney classes of the fixed bundle.

Keywords. Cohomology rings, equivariant cohomology, spaces with involution, real spaces

AMS subject classification. Primary: 55N91, 55M35. Secondary: 53D05, 57R22.

E-print: arXiv:math.AT/0412057

DOI: 10.2140/agt.2005.5.923

Submitted: 16 February 2005. Accepted: 7 July 2005. Published: 5 August 2005.

Notes on file formats

Jean-Claude Hausmann, Tara Holm and Volker Puppe

Section de mathematiques, 2-4, rue du Lievre
CP 64 CH-1211 Geneve 4, Switzerland

Department of Mathematics, University of Connecticut
Storrs CT 06269-3009, USA

Universitat Konstanz, Fakultat fur Mathematik
Fach D202, D-78457 Konstanz, Germany


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