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 $\mathbb {R} \mathbb {P} ^{2}$	1	homology of real projective space
even-dimensional real projective space $\mathbb {R} \mathbb {P} ^{2m}$	1	homology of real projective space
odd-dimensional real projective space $\mathbb {R} \mathbb {P} ^{n}$, $n=2m+1$	$1+x^{n}$	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:

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