User:EverettYou/Poincaré polynomial

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).