Poincaré transversality for double covers

Let π: X’ —” X be a double cover of 2n-dimensional Poincaré duality (PD) spaces. The double cover is a fibering so it is classified by a map f: X → RP1+1(l ≫ n). If the homotopy class of f contains a representative which is Poincaré transverse [5] to RP l ⊂ RP l+1 , we say that w is Poincaré split-table.