#### Volume 5, issue 3 (2005)

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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 $\tau$ whose mod 2 cohomology ring resembles that of their fixed point set ${X}^{\tau }$: there is a ring isomorphism $\kappa :\phantom{\rule{0.3em}{0ex}}{H}^{2\ast }\left(X\right)\approx {H}^{\ast }\left({X}^{\tau }\right)$. 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 ${H}^{\ast }$–frame. An ${H}^{\ast }$–frame, if it exists, is natural and unique. A space with involution admitting an ${H}^{\ast }$–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
##### Mathematical Subject Classification 2000
Primary: 55N91, 55M35
Secondary: 53D05, 57R22