# 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.