Rational relative toral rank

For a map f : X → Y between simply connected CW complexes with finite rational cohomologies, we define the rational toral rank of X respect to f , r 0 ( X , f ) . It is a generalization of the rational toral rank r 0 ( X ) (Halperin in Rational Homotopy and Torus Actions, Cambridge University Press, Cambridge, 1985) of X . Then we note some properties of them by using Sullivan minimal models. Furthermore we propose a relative Halperin conjecture.