#### 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

Email: hausmann@math.unige.ch, tsh@math.uconn.edu, Volker.Puppe@uni-konstanz.de

AGT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**