User:EverettYou/Poincaré polynomial

Definition
Given a topological space X which has finitely generated homology, the Poincaré polynomial of X, denoted as P(X), is defined as the generating function of its Betti numbers bp,
 * $$P(X)=b_0(X)+b_1(X)x+b_2(X)x^2+\cdots=\sum_p b_p(X)x^p.$$

For infinite-dimensional spaces, the Poincaré polynomial is generalized to Poincaré series.

Table of Poincaré polynomials
The Poincaré polynomials of the compact simple Lie groups.

Disjoint Union
Let $$X \sqcup Y$$ be the disjoint union of spaces X and Y.
 * $$P(X \sqcup Y) = P(X) + P(Y).$$

Wedge Sum
Let $$X \vee Y$$ be the wedge sum of two path-connected spaces X and Y.
 * $$P(X \vee Y) = P(X) + P(Y) - 1.$$

Connected Sum
If X and Y are compact connected manifolds of the same dimension n, then the Poincaré polynomial of their connected sum X#Y is
 * $$P(X\#Y) = P(X) + P(Y) - P(S^n).$$

Product
The Poincaré polynomial of the product of the spaces X×Y is
 * $$P(X \times Y) = P(X) \times P(Y).$$

This is a corollary of the Kunneth formula (note that we are assuming that both spaces have finitely generated homology).