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User:EverettYou/Poincaré polynomial

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Definition

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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 bp,

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

Table of Poincaré polynomials

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disk 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

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Disjoint Union

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Let be the disjoint union of spaces X and Y.

Wedge Sum

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Let be the wedge sum of two path-connected spaces X and Y.

Connected Sum

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If X and Y are compact connected manifolds of the same dimension n, then the Poincaré polynomial of their connected sum X#Y is

Product

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The 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).