Poincare polynomial of product is product of Poincare polynomials

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Statement

For two spaces with finitely generated homology, over the integers

Suppose and are (possibly homeomorphic/equal) topological spaces and both have finitely generated homology over the integers, i.e., at most finitely many of the homology groups of are nonzero, and all these are finitely generated, and the same holds for . In particular, this means that the Poincare polynomial (?)s and are defined.

Then, the Cartesian product , equipped with the product topology, is also a space with finitely generated homology over the integers, and its Poincare polynomial is given by:

where the multiplication on the right is as multiplication of polynomials in .

For finitely many spaces with finitely generated homology, over the integers

Suppose are all (possibly homeomorphic/equal) topological spaces, each of which has finitely generated homology over the integers. Then, the Cartesian product is also a topological space with finitely generated homology, and its Poincare polynomial is the product of the Poincare polynomials of each of the s, i.e.:

where the multiplication on the right is carried out as polynomials in .

Over an arbitrary commutative unital ring

Analogous statements to the above hold if we replace the ring of integers by an arbitrary commutative unital ring.

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