This article describes an invariant of topological spaces that depends only on its homology groups
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 .
The Poincare polynomial of is denoted .
|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|
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:
The Poincare polynomial of a wedge sum of two path-connected spaces, is the sum of their polynomials minus 1.
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: