# Poincare polynomial

*This article describes an invariant of topological spaces that depends only on its homology groups*

## Definition

Given a topological space which has finitely generated homology, the Poincare polynomial of is defined as the generating function of its Betti numbers, viz the polynomial where the coefficient of is .

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 is denoted .

## 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 | homology of spheres | |

torus (product of copies of the circle) | homology of torus | |

sphere | homology of spheres | |

product of spheres | homology of product of spheres | |

compact orientable genus surface | homology of compact orientable surfaces | |

real projective plane | 1 | homology of real projective space |

even-dimensional real projective space | 1 | homology of real projective space |

odd-dimensional real projective space , | homology of real projective space | |

complex projective space | 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:

### Wedge sum

The Poincare polynomial of a wedge sum of two path-connected spaces, is the sum of their polynomials minus 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: