Steenrod squares on conjugation spaces

Let be an involution on a topological spaceX, that is, an action onX of the cyclic group of order two G = {1, } whose fixed point set isX . LetH denote singular cohomology with coefficients in Z/2Z andH G itsG-equivariant version. Let r:H (X)!H (X ) and rG:H G (X)!H G (X ) be the restriction maps and p:H G (X)!H (X) the canonical projection. Recall that H G (X ) is isomorphic to the polynomial ring H (X )[u], where u has degree one. A conjugation space (X, ) is a topological spaceX with an involution such thatHodd(X) = 0 and there exist a degree-halving isomorphism k:H2 (X)!H (X ) and a section :H2 (X)! H2 G (X) of p such that, for every x 2 H2n(X), the conjugation equation rG( (x)) = k(x)un + q(u) holds, where q(u) is a polynomial in H (X )[u] of degree less than n. Flag manifolds, co-adjoint orbits of compact Lie groups and compact toric manifolds are all examples of conjugation spaces. This paper contains an explicit expression for the conjugation equation for any conjugation space; it shows that the coefficients of the uis are completely determined by the Steenrod squares. It also generalizes a result of V. A. Krasnov [Mat. Zametki 73 (2003), no. 6, 853–860; MR2010655 (2004k:14111)], and a formula by A. Borel and A. Haefliger [Bull. Soc. Math. France 89 (1961), 461–513; MR0149503 (26 #6990)] concerning a class of complex projective varieties.