The tangential end fibration of an aspherical Poincaré complex

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