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 b_p,

$P(X)=b_{0}(X)+b_{1}(X)x+b_{2}(X)x^{2}+\cdots =\sum _{p}b_{p}(X)x^{p}.$

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

Table of Poincaré polynomials

$X$	$P(X)$
disk Dn	1
circle S1	$1+x$
sphere Sn	$1+x^{n}$
torus Tn	$(1+x)^{n}$
genus g surface $T_{g}^{2}$	$1+2gx+x^{2}$
real space $\mathbb {R} ^{n}$	1
$\mathbb {R} P^{2m}$	1
$\mathbb {R} P^{2m+1}$	$1+x^{2m+1}$
$\mathbb {C} P^{n}$	$1+x^{2}+\cdots +x^{2n}$

The Poincaré polynomials of the compact simple Lie groups.

$X$	$P(X)$
SU(n+1)	$(1+x^{3})(1+x^{5})...(1+x^{2n+1})$
SO(2n+1)	$(1+x^{3})(1+x^{7})...(1+x^{4n-1})$
SO(2n)	$(1+x)(1+x^{3})^{2}...(1+x^{2n-1})(1+x^{4n-5})$
Sp(2n)	$(1+x^{3})(1+x^{7})...(1+x^{4n-1})$
G2	$(1+x^{3})(1+x^{11})$
F4	$(1+x^{3})(1+x^{11})(1+x^{15})(1+x^{23})$
E6	$(1+x^{3})(1+x^{9})(1+x^{11})(1+x^{15})(1+x^{17})(1+x^{23})$
E7	$(1+x^{3})(1+x^{11})(1+x^{15})(1+x^{19})(1+x^{23})(1+x^{27})(1+x^{35})$
E8	$(1+x^{3})(1+x^{15})(1+x^{23})(1+x^{27})(1+x^{35})(1+x^{39})(1+x^{47})(1+x^{59})$

Formulae of Poincaré Polynomial

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).