Algebraic & Geometric Topology

Conjugation spaces

Jean-Claude Hausmann, Tara S Holm, and Volker Puppe

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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.

Article information

Algebr. Geom. Topol., Volume 5, Number 3 (2005), 923-964.

Received: 16 February 2005
Accepted: 7 July 2005
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 55N91: Equivariant homology and cohomology [See also 19L47] 55M35: Finite groups of transformations (including Smith theory) [See also 57S17]
Secondary: 53D05: Symplectic manifolds, general 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx]

cohomology rings equivariant cohomology spaces with involution real spaces


Hausmann, Jean-Claude; Holm, Tara S; Puppe, Volker. Conjugation spaces. Algebr. Geom. Topol. 5 (2005), no. 3, 923--964. doi:10.2140/agt.2005.5.923.

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