Poincare polynomial

This article describes an invariant of topological spaces that depends only on its homology groups

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 $PX$.

Facts

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

When either of the spaces has torsion-free homology, 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:

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