Poincare polynomial of fiber product need not equal product of Poincare polynomials

Statement
It is possible to have a fiber product $$E$$ with base space $$B$$ and fiber space $$F$$, such that all of them are spaces with finitely generated homology, and the fact about::Poincare polynomial of $$E$$ is not the product of the Poincare polynomials of $$B$$ and $$F$$.

In fact, we can choose an example where all the spaces involved are fact about::manifolds.

Related facts

 * Poincare polynomial of product is product of Poincare polynomials
 * Euler characteristic of product is product of Euler characteristics

Proof
Consider the example where $$E$$ is the Klein bottle and $$B$$ and $$F$$ are both circles.

In this case, the Poincare polynomials for $$B$$, $$F$$, and $$E$$ are all equal to $$1 + x$$.