Poincare polynomial

Definition
Given a topological space $$X$$ which has finitely generated homology, the Poincare polynomial of $$X$$ is defined as the generating function of its Betti numbers, viz the polynomial where the coefficient of $$x^q$$ is $$b_q(X)$$.

Note that for a space with homology of finite type, all the Betti numbers are well-defined, but infinitely many of them are nonzero, so we get a Poincare series instead of a Poincare polynomial.

The Poincare polynomial of $$X$$ is denoted $$PX$$.

Disjoint union
The Poincare polynomial of a disjoint is the sum of the Poincare polynomials of the individual spaces:

$$P(X \sqcup Y) = PX + PY$$

Wedge sum
The Poincare polynomial of a wedge sum of two path-connected spaces, is the sum of their polynomials minus 1.

$$P(X \vee Y) = PX + PY - 1$$

Product
The Poincare polynomial of the product of the spaces is the product of the Poincare polynomials. This is a corollary of the Kunneth formula (note that we are assuming that both spaces have finitely generated homology).

A particular case of this (which can be proved directly using the exact sequence for join and product and does not require appeal to the Kunneth formula) is:

$$P(X \times S^m) = PX \times P(S^m)$$