Poincare polynomial: Difference between revisions

Revision as of 21:45, 3 November 2007

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

Definition

Given a topological space $X$ which has finitely generated homology, the Poincare polynomial of $X$ is defined as the generating function of its Betti numbers, viz the polynomial where the coefficient of $x^{q}$ is $b_{q}(X)$ .

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 $X$ is denoted $PX$ .

Facts

Disjoint union

The Poincare polynomial of a disjoint is the sum of the Poincare polynomials of the individual spaces:

$P(X\sqcup Y)=PX+PY$

Wedge sum

The Poincare polynomial of a wedge sum of two path-connected spaces, is the sum of their polynomials minus 1.

$P(X\vee Y)=PX+PY-1$

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.

$P(X\times S^{m})=PX\times P(S^{m})$

@@ Line 14: / Line 14: @@
 ===Disjoint union===
-The Poincare polynomial of a disjoint is the sum of the Poincare polynomials of the individual spaces.
+The Poincare polynomial of a disjoint is the sum of the Poincare polynomials of the individual spaces:
+<math>P(X \sqcup Y) = PX + PY</math>
 ===Wedge sum===
 The Poincare polynomial of a wedge sum of two path-connected spaces, is the sum of their polynomials minus 1.
+<math>P(X \vee Y) = PX + PY  - 1</math>
 ===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.
+<math>P(X \times S^m) = PX \times P(S^m)</math>