Is there a formula for the surgery obstruction of a fiber product of maps assuming the fiber product is also a homotopy equivalence?
In more detail, suppose that $X \to Y$ and $Z \to Y$ are homotopy equivalences of (differentiable, say, compact oriented) manifolds. There is a long history of product formulae for surgery obstructions which includes a surgery obstruction for the product map $X \times Z \to Y \times Y$ to be homotopic to a homeomorphism. See for example Section 8 in Ranicki, Algebraic Theory of Surgery II. Applications to Topology.
Suppose I know that the fiber product $X \times_Y Z \to Y$ is also a homotopy equivalence. (Often it won't be, but in my case I got lucky.) Is there a formula for the surgery obstruction of this map in terms of those of $X \to Y$ and $Z \to Y$? I am actually just interested in the case $X \times_Y X \to Y$, perturbed to make the fiber product transverse and would like to know what are the possibilities for the surgery obstruction of this map. (Could it be anything?)
Side question: It's interesting to compare the honest fiber product $X \times_Y Z $ with the homotopy fiber product $X \times_{Y,h} Z$; in my case they are homotopy equivalent. But I don't see any reason why $X \times_{Y,h} Z$ should admit a manifold structure in general. That is, Given $X,Y,Z$ homotopy equivalent as above, when is should one expect the homotopy fiber product $X \times_{Y,h} Z$ to admit a manifold structure?