Volume 5, issue 3 (2005)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 6, 2971–3570
Issue 5, 2389–2970
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
Conjugation spaces

Jean-Claude Hausmann, Tara S Holm and Volker Puppe

Algebraic & Geometric Topology 5 (2005) 923–964

arXiv: math.AT/0412057

Abstract

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.

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