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 $PX$.

Particular cases

Case for the space Value of Poincare polynomial (note: if the space is not a single space but a parameterized family of spaces, the polynomial may also depend on the parameter) Further information
a contractible space 1
circle $1 + x$ homology of spheres
torus $T^n$ (product of $n$ copies of the circle) $(1 + x)^n$ homology of torus
sphere $S^n$ $1 + x^n$ homology of spheres
product of spheres $S^{m_1} \times S^{m_2} \times \dots \times S^{m_r}$ $(1 + x^{m_1})(1 + x^{m_2}) \dots (1 + x^{m_r})$ homology of product of spheres
compact orientable genus $g$ surface $1 + 2gx + x^2$ homology of compact orientable surfaces
real projective plane $\R\mathbb{P}^2$ 1 homology of real projective space
even-dimensional real projective space $\R\mathbb{P}^{2m}$ 1 homology of real projective space
odd-dimensional real projective space $\R\mathbb{P}^n$, $n = 2m + 1$ $1 + x^n$ homology of real projective space
complex projective space $\mathbb{C}\mathbb{P}^n$ $1 + x^2 + \dots + x^{2n}$ homology of complex projective space

Facts

Disjoint union

Further information: Poincare polynomial of disjoint union is sum of Poincare polynomials

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

Further information: Poincare polynomial of product is product of Poincare polynomials

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:

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