Here when there are finitely many Betti numbers of "X" and "Y", each of which is a natural number rather than formula_11, this reads as an identity on Poincaré polynomials. In the general case these are formal power series with possibly infinite coefficients, and have to be interpreted accordingly. Furthermore, the above statement holds not only for the Betti numbers but also for the generating functions of the dimensions of the homology over any field. (If the integer homology is not torsion-free, then these numbers may differ from the standard Betti numbers.)