Doubling Operation for Polytopes and Torus Actions

In this note we give the definition of the ”doubling operation” for simple polytopes, find the formula for the h-polynomial of new polytope. As an application of this operation we establish the relationship between moment-angle manifolds and their real analogues and prove the toral rank conjecture for moment-angle manifolds ZP . Let P be a simple n-dimensional polytope with m facets: P = {x ∈ R | (x, ai) + bi > 0, ai ∈ R, i = 1 . . .m}. Then P can be identified with the intersection of the positive orthant Rm> and the image of the affine map iP : R n → R, iP (x) = AP (x) + bP , where AP is m × n matrix with rows ai and bP is column vector (b1, . . . , bm). Suppose the image of the map iP is specified by the system of m− n equations in R :