## Algebraic & Geometric Topology

### Conjugation spaces

#### 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 $ℂk$ 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

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

Dates
Accepted: 7 July 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796440

Digital Object Identifier
doi:10.2140/agt.2005.5.923

Mathematical Reviews number (MathSciNet)
MR2171799

Zentralblatt MATH identifier
1081.55006

#### Citation

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. https://projecteuclid.org/euclid.agt/1513796440

#### References

• C Allday, V Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics 32, Cambridge University Press, Cambridge (1993)\relax
• M F Atiyah, $K$-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966) 367–386\relax
• M Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics 93, Birkhäuser Verlag, Basel (1991)\relax
• D Biss, V W Guillemin, T S Holm, The mod 2 cohomology of fixed point sets of anti-symplectic involutions, Adv. Math. 185 (2004) 370–399\relax
• A Borel, Seminar on transformation groups, with contributions by G. Bredon, E E Floyd, D Montgomery, R Palais, Annals of Mathematics Studies 46, Princeton University Press, Princeton, N.J. (1960)\relax
• A Borel, A Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89 (1961) 461–513\relax
• M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417–451\relax
• J J Duistermaat, Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution, Trans. Amer. Math. Soc. 275 (1983) 417–429\relax
• R F Goldin, T S Holm, Real loci of symplectic reductions, Trans. Amer. Math. Soc. 356 (2004) 4623–4642\relax
• M Goresky, R Kottwitz, R MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998) 25–83\relax
• M J Greenberg, J R Harper, Algebraic topology, Mathematics Lecture Note Series 58, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass. (1981)\relax
• T Holm, M Harada, The equivariant cohomology of hypertoric varieties and their real loci, Commun. Anal. Geom. 13 (2005) 645–677
• J-C Hausmann, A Knutson, The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998) 281–321\relax
• J-C Hausmann, S Tolman, Maximal Hamiltonian tori for polygon spaces, Ann. Inst. Fourier (Grenoble) 53 (2003) 1925–1939\relax
• D Husemoller, Fibre bundles, Springer-Verlag, New York (1975)\relax
• F C Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes 31, Princeton University Press, Princeton, NJ (1984)\relax
• E Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995) 247–258\relax
• A T Lundell, S Weingram, The topology of CW-complexes Van Nostrand (1969)
• A A Klyachko, Spatial polygons and stable configurations of points in the projective line, from: “Algebraic geometry and its applications (Yaroslavl', 1992)”, Aspects Math. E25, Vieweg, Braunschweig (1994) 67–84\relax
• J McCleary, A user's guide to spectral sequences, second edition, Cambridge Studies in Advanced Mathematics 58, Cambridge University Press (2001)
• J W Milnor, J D Stasheff, Characteristic classes, Princeton University Press, Princeton, NJ (1974)\relax
• L O'Shea, R Sjamaar, Moment maps and Riemannian symmetric pairs, Math. Ann. 317 (2000) 415–457\relax
• H Samelson, Notes on Lie algebras, Universitext, Springer-Verlag, New York (1990)\relax
• C Schmid, Cohomologie équivariante de certaines variétés hamiltoniennes et de leur partie réelle, thesis, University of Geneva (2001)
• E H Spanier, Algebraic topology, McGraw-Hill Book Co. New York (1966)\relax
• T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co. Berlin (1987)\relax
• S Tolman, J Weitsman, The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 (2003) 751–773\relax
• J A van Hamel, Algebraic cycles and topology of real algebraic varieties, CWI Tract 129, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam (2000)\relax
• J A van Hamel, Personal correspondence, (March 2004)