Definition
editGiven 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 bp,
For infinite-dimensional spaces, the Poincaré polynomial is generalized to Poincaré series.
Table of Poincaré polynomials
editdisk Dn | 1 |
circle S1 | |
sphere Sn | |
torus Tn | |
genus g surface | |
real space | 1 |
1 | |
The Poincaré polynomials of the compact simple Lie groups.
SU(n+1) | |
SO(2n+1) | |
SO(2n) | |
Sp(2n) | |
G2 | |
F4 | |
E6 | |
E7 | |
E8 |
Formulae of Poincaré Polynomial
editDisjoint Union
editLet be the disjoint union of spaces X and Y.
Wedge Sum
editLet be the wedge sum of two path-connected spaces X and Y.
Connected Sum
editIf X and Y are compact connected manifolds of the same dimension n, then the Poincaré polynomial of their connected sum X#Y is
Product
editThe Poincaré polynomial of the product of the spaces X×Y is
This is a corollary of the Kunneth formula (note that we are assuming that both spaces have finitely generated homology).