Homology of real projective space: Difference between revisions

Revision as of 18:55, 31 December 2010

This article describes the homology of the following space or class of spaces: real projective space

Statement

Odd-dimensional projective space with coefficients in integers

$H_{p} (P^{n} (R)) = Z p = 0, n$

For odd $p$ with $0 < p < n$ :

$H_{p} (P^{n} (R)) = Z / 2 Z$

And zero otherwise.

Even-dimensional projective space with coefficients in integers

We have:

$H_{p} (P^{n} (R)) = Z p = 0$

For odd $p$ with $0 < p < n$ :

$H_{p} (P^{n} (R)) = Z / 2 Z$

And zero otherwise.

Thus the key difference between even and odd dimensional projective spaces is that the top homology vanishes in even-dimensional projective spaces. This is related to the fact that even-dimensional projective space is non-orientable, while odd-dimensional projective space is orientable.

Coefficients in a 2-divisible ring

If we consider the homology with coefficients in a module $M$ over a ring $R$ where 2 is invertible, the we have: $H_{n} (P^{n} (R); M) ≅ H_{0} (P^{n} (R); M) ≅ M$ (regardless of whether $n$ is even or odd) and all other homology groups are zero.

In particular, these results are valid over the field of rational numbers or over any field of characteristic zero.

Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant	General description	Description of value for spheres
Betti numbers	The $n^{t h}$ Betti number $b_{k}$ is the rank of the $k^{t h}$ homology group.	$b_{0} = 1$ . $b_{n} = 1$ if $n$ is odd and $b_{n} = 0$ if $n$ is even.
Poincare polynomial	Generating polynomial for Betti numbers	$1 + x^{n}$ if $n$ is odd. $1$ if $n$ is even.
Euler characteristic	$\sum_{k = 0}^{\infty} (- 1)^{k} b_{k}$	$0$ if $n$ is odd. $1$ if $n$ is even. Note that the Euler characteristic is half the Euler characteristic of $S^{n}$ , which is its double cover.

Facts used

CW structure of real projective space

Proof

The proof follows from fact (1). See more details on that page.

@@ Line 3: / Line 3: @@
 ==Statement==
-===Even-dimensional projective space===
+===Odd-dimensional projective space with coefficients in integers===
 <math>H_p(\mathbb{P}^n(\R)) = \Z \qquad p=0,n</math>
@@ Line 13: / Line 13: @@
 And zero otherwise.
-===Odd-dimensional projective space===
+===Even-dimensional projective space with coefficients in integers===
 We have:
@@ Line 27: / Line 27: @@
 Thus the key difference between even and odd dimensional projective spaces is that the top homology vanishes in even-dimensional projective spaces. This is related to the fact that even-dimensional projective space is non-orientable, while odd-dimensional projective space is orientable.
-==Related invariants==
+===Coefficients in a 2-divisible ring===
+If we consider the homology with coefficients in a module <math>M</math> over a ring <math>R</math> where 2 is invertible, the we have: <math>H_n(\mathbb{P}^n(\R);M) \cong H_0(\mathbb{P}^n(\R);M) \cong M</math> (regardless of whether <math>n</math> is even or odd) and all other homology groups are zero.
-===Betti numbers===
+In particular, these results are valid over the field of rational numbers or over any field of characteristic zero.
-The Betti numbers of real projective space are thus <math>1</math> at <math>0</math> and at <math>n</math> if <math>n</math> is odd, and <math>1</math> only at <math>0</math> if <math>n</math> is even.
+==Related invariants==
-===Poincare polynomial===
+These are all invariants that can be computed in terms of the homology groups.
-The Poincare polynomial of real projective space is <math>1 + x^n</math> if <math>n</math> is odd, and <math>1</math> if <math>n</math> is even.
+{| class="sortable" border="1"
+! Invariant !! General description !! Description of value for spheres
+|-
+| [[Betti number]]s || The <math>n^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. || <math>b_0 = 1</math>. <math>b_n = 1</math> if <math>n</math> is odd and <math>b_n = 0</math> if <math>n</math> is even.
+|-
+| [[Poincare polynomial]] || Generating polynomial for Betti numbers || <math>1 + x^n</math> if <math>n</math> is odd. <math>1</math> if <math>n</math> is even.
+|-
+| [[Euler characteristic]] || <math>\sum_{k=0}^\infty (-1)^k b_k</math> || <math>0</math> if <math>n</math> is odd. <math>1</math> if <math>n</math> is even. Note that the Euler characteristic is half the Euler characteristic of <math>S^n</math>, which is its double cover.
+|}
-===Euler characteristic===
+==Facts used==
-The Euler characteristic is 0 if <math>n</math> is odd and <math>1</math> if <math>n</math> is even.
+# [[uses::CW structure of real projective space]]
-==Relation with the sphere==
+==Proof==
-There is a double cover from the <math>n</math>-sphere to real projective <math>n</math>-space. This double cover induces an isomorphism on all even-dimensional homologies (and of course on all homologies higher than <math>n</math>). {{fillin}}
+The proof follows from fact (1). See more details on that page.