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 $P X$ .

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	$1 + 2 g x + x^{2}$	homology of compact orientable surfaces
real projective plane $R P^{2}$	1	homology of real projective space
even-dimensional real projective space $R P^{2 m}$	1	homology of real projective space
odd-dimensional real projective space $R P^{n}$ , $n = 2 m + 1$	$1 + x^{n}$	homology of real projective space
complex projective space $C P^{n}$	$1 + x^{2} + \dots + x^{2 n}$	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:

$P (X ⊔ Y) = P X + P Y$

Wedge sum

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

$P (X \lor Y) = P X + P Y - 1$

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:

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