The tangential end fibration of an aspherical Poincaré complex

We construct a sphere fibration over a finite aspherical Poincaré complex X , which we call the tangential end fibration, under the condition that the universal cover of X is forward tame and simply connected at infinity. We show that it is tangent to X if the formal dimension of X is even or, when the formal dimension is odd, if the diagonal X → X × X admits a Poincaré embedding structure.