Poincare polynomial: Difference between revisions

Latest revision as of 00:09, 2 April 2011

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 $PX$ .

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	$1+x$	homology of spheres
torus $T^{n}$ (product of $n$ copies of the circle)	$(1+x)^{n}$	homology of torus
sphere $S^{n}$	$1+x^{n}$	homology of spheres
product of spheres $S^{m_{1}}\times S^{m_{2}}\times \dots \times S^{m_{r}}$	$(1+x^{m_{1}})(1+x^{m_{2}})\dots (1+x^{m_{r}})$	homology of product of spheres
compact orientable genus $g$ surface	$1+2gx+x^{2}$	homology of compact orientable surfaces
real projective plane $\mathbb {R} \mathbb {P} ^{2}$	1	homology of real projective space
even-dimensional real projective space $\mathbb {R} \mathbb {P} ^{2m}$	1	homology of real projective space
odd-dimensional real projective space $\mathbb {R} \mathbb {P} ^{n}$ , $n=2m+1$	$1+x^{n}$	homology of real projective space
complex projective space $\mathbb {C} \mathbb {P} ^{n}$	$1+x^{2}+\dots +x^{2n}$	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:

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

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

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:

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

@@ Line 9: / Line 9: @@
 The Poincare polynomial of <math>X</math> is denoted <math>PX</math>.
+==Particular cases==
+{| class="sortable" border="1"
+! 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]] || <math>1 + x</math> || [[homology of spheres]]
+|-
+| [[torus]] <math>T^n</math> (product of <math>n</math> copies of the circle) || <math>(1 + x)^n</math> || [[homology of torus]]
+|-
+| [[sphere]] <math>S^n</math> || <math>1 + x^n</math> || [[homology of spheres]]
+|-
+| [[product of spheres]] <math>S^{m_1} \times S^{m_2} \times \dots \times S^{m_r}</math> || <math>(1 + x^{m_1})(1 + x^{m_2}) \dots (1 + x^{m_r})</math> || [[homology of product of spheres]]
+|-
+| compact orientable genus <math>g</math> surface || <math>1 + 2gx + x^2</math> || [[homology of compact orientable surfaces]]
+|-
+| [[real projective plane]] <math>\R\mathbb{P}^2</math> || 1 || [[homology of real projective space]]
+|-
+| even-dimensional [[real projective space]] <math>\R\mathbb{P}^{2m}</math> || 1 || [[homology of real projective space]]
+|-
+| odd-dimensional real projective space <math>\R\mathbb{P}^n</math>, <math>n = 2m + 1</math> || <math>1 + x^n</math> || [[homology of real projective space]]
+|-
+| [[complex projective space]] <math>\mathbb{C}\mathbb{P}^n</math> || <math>1 + x^2 + \dots + x^{2n}</math> || [[homology of complex projective space]]
+|}
 ==Facts==
@@ Line 14: / Line 39: @@
 ===Disjoint union===
-The Poincare polynomial of a disjoint is the sum of the Poincare polynomials of the individual spaces.
+{{further|[[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:
+<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.
+{{further|[[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:
+<math>P(X \times S^m) = PX \times P(S^m)</math>