University of Toronto's Symplectic Geometry Seminar

November 15, 2004, 13:10--14:00
SS 5017A

Jean-Claude Hausmann

University of Geneva

Conjugation spaces and bundles

Abstract: There are classical examples of spaces X with an involution \tau whose mod 2-comhomology ring ressembles 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, etx. In this talk, we show that, for the above examples and many others, 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^*frames is called a {\it conjugation space}. Many examples of conjugation spaces can be constructed. Among them, one has the compact symplectic manifolds, when they admit an anti-symplectic involution compatible with a Hamiltonian action of a torus T, provided X^T is a itself a conjugation space (e.g. if X^T is discrete). We will aslo study a couple of bundles over conjugation spaces. For a conjugate-equivariant complex vector bundle (``real bundle'' in the sense of Atiyah), we show that the isomorphism \kappa sends its Chern classes onto the Stiefel-Whitney classes of its fixed bundle.