Conjugation spaces. (English) Zbl 1081.55006

This paper defines the notion of conjugation space and studies the properties of such spaces. These are spaces \(X\) having an involution \(T\) for which \(X\) and the fixed point set \(F\) of \(T\) have isomorphic mod 2 cohomology with dimensions doubled. The classic example is a complex Grassmannian with involution given by complex conjugation for which the fixed point set is a real Grassmannian. Many examples of conjugation spaces are exhibited.


55N91 Equivariant homology and cohomology in algebraic topology
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
53D05 Symplectic manifolds (general theory)
57R22 Topology of vector bundles and fiber bundles
Full Text: DOI arXiv EuDML EMIS


[1] C Allday, V Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics 32, Cambridge University Press (1993) · Zbl 0799.55001
[2] M F Atiyah, \(K\)-theory and reality, Quart. J. Math. Oxford Ser. \((2)\) 17 (1966) 367 · Zbl 0146.19101
[3] M Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics 93, Birkhäuser Verlag (1991) 181 · Zbl 0726.57029
[4] 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 · Zbl 1069.53058
[5] A Borel, Seminar on transformation groups, With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics Studies, No. 46, Princeton University Press (1960)
[6] A Borel, A Haefliger, La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961) 461 · Zbl 0102.38502
[7] M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417 · Zbl 0733.52006
[8] 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 · Zbl 0504.58020
[9] R F Goldin, T S Holm, Real loci of symplectic reductions, Trans. Amer. Math. Soc. 356 (2004) 4623 · Zbl 1054.53096
[10] M Goresky, R Kottwitz, R MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998) 25 · Zbl 0897.22009
[11] M J Greenberg, J R Harper, Algebraic topology, Mathematics Lecture Note Series 58, Benjamin/Cummings Publishing Co. Advanced Book Program (1981) · Zbl 0498.55001
[12] M Harada, T S Holm, The equivariant cohomology of hypertoric varieties and their real loci, Comm. Anal. Geom. 13 (2005) 527 · Zbl 1088.53055
[13] J C Hausmann, A Knutson, The cohomology ring of polygon spaces, Ann. Inst. Fourier (Grenoble) 48 (1998) 281 · Zbl 0903.14019
[14] J C Hausmann, S Tolman, Maximal Hamiltonian tori for polygon spaces, Ann. Inst. Fourier (Grenoble) 53 (2003) 1925 · Zbl 1049.53056
[15] D Husemoller, Fibre bundles, Graduate Texts in Mathematics 20, Springer (1975) · Zbl 0307.55015
[16] F C Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes 31, Princeton University Press (1984) · Zbl 0553.14020
[17] E Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995) 247 · Zbl 0835.53034
[18] A T Lundell, S Weingram, The topology of CW-complexes, Van Nostrand (1969) · Zbl 0207.21704
[19] A A Klyachko, Spatial polygons and stable configurations of points in the projective line, Aspects Math., E25, Vieweg (1994) 67 · Zbl 0820.51016
[20] J McCleary, A user’s guide to spectral sequences, Cambridge Studies in Advanced Mathematics 58, Cambridge University Press (2001) · Zbl 0959.55001
[21] J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press (1974) · Zbl 0298.57008
[22] L O’Shea, R Sjamaar, Moment maps and Riemannian symmetric pairs, Math. Ann. 317 (2000) 415 · Zbl 0985.37056
[23] H Samelson, Notes on Lie algebras, Universitext, Springer (1990) · Zbl 0708.17005
[24] C Schmid, Cohomologie équivariante de certaines variétés hamiltoniennes et de leur partie réelle, thesis, University of Geneva (2001) · Zbl 1026.55008
[25] E H Spanier, Algebraic topology, McGraw-Hill Book Co. (1966) · Zbl 0145.43303
[26] T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co. (1987) · Zbl 0611.57002
[27] S Tolman, J Weitsman, The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 (2003) 751 · Zbl 1087.53076
[28] J A van Hamel, Algebraic cycles and topology of real algebraic varieties, CWI Tract 129, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica (2000) · Zbl 0986.14042
[29] J A van Hamel, personal correspondence (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.