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
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