Editing Betti number (section)

== Poincaré polynomial ==
The '''Poincaré polynomial''' of a surface is defined to be the [[generating function]] of its Betti numbers.  For example, the Betti numbers of the torus are 1, 2, and 1; thus its Poincaré polynomial is <math>1+2x+x^2</math>. The same definition applies to any topological space  which has a finitely generated homology.

Given a topological space which has a finitely generated homology, the Poincaré polynomial is defined as the generating function of its Betti numbers, via the polynomial where the coefficient of <math>x^n</math> is <math>b_n</math>.