Poincare polynomial: Difference between revisions
No edit summary |
No edit summary |
||
| Line 3: | Line 3: | ||
==Definition== | ==Definition== | ||
Given a [[topological space]] <math>X</math> which has [[space | Given a [[topological space]] <math>X</math> which has [[space with finitely generated homology|finitely generated homology]], the Poincare polynomial of <math>X</math> is defined as the generating function of its Betti numbers, viz the polynomial where the coefficient of <math>x^q</math> is <math>b_q(X)</math>. | ||
Note that for a [[space 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. | 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. | ||
==Facts== | ==Facts== | ||
Revision as of 21:43, 3 November 2007
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.
Facts
Disjoint union
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
When either of the spaces is a sphere, the Poincare polynomial of the product of the spaces is the product of the Poincare polynomials. However, the result does not hold for arbitrary topological spaces.