Conjugation spaces

There are classical examples of spaces $X$ with an involution $τ$ whose mod 2 cohomology ring resembles that of their fixed point set $X^{τ}$ : there is a ring isomorphism $κ : H^{2 *} (X) \approx H^{*} (X^{τ})$ . 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 $ℂ^{k}$ 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 $κ$ maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.

Keywords

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

Mathematical Subject Classification 2000

Primary: 55N91, 55M35

Secondary: 53D05, 57R22

References

Forward citations

Publication

Received: 16 February 2005

Accepted: 7 July 2005

Published: 5 August 2005

Authors

Jean-Claude Hausmann
Section de mathématiques 2–4 rue du Lièvre CP 64 CH-1211 Genève 4 Switzerland

Tara S Holm
Department of Mathematics University of Connecticut Storrs CT 06269-3009 USA

Volker Puppe
Universität Konstanz Fakultät für Mathematik Fach D202 D-78457 Konstanz Germany

Volume 5, issue 3 (2005)

Jean-Claude Hausmann, Tara S Holm and Volker Puppe

Abstract