Cohomology of real projective space: Difference between revisions

Revision as of 04:39, 26 July 2011

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology group and the topological space/family is real projective space
Get more specific information about real projective space | Get more computations of homology group

Statement

Odd-dimensional projective space with coefficients in integers

$H_{p} (P^{n} (R)) = {\begin{matrix} Z, & p = 0, n \\ Z / 2 Z, & p e v e n, 0 < p < n \\ 0, & o t h e r w i s e \end{matrix}$

Even-dimensional projective space with coefficients in integers

$H_{p} (P^{n} (R)) = {\begin{matrix} Z, & p = 0 \\ Z / 2 Z, & p e v e n, 0 < p \leq n \\ 0, & o t h e r w i s e \end{matrix}$

Cohomology groups with integer coefficients in tabular form

We illustrate how the cohomology groups work for small values of $n$ . Note that for $p > n$ , all cohomology groups $H^{p}$ are zero, so we omit those cells for visual ease.

$n$	Real projective space $R P^{n}$	Orientable?	$H_{0}$	$H_{1}$	$H_{2}$	$H_{3}$	$H_{4}$	$H_{5}$
1	circle	Yes	$Z$	$Z$
2	real projective plane	No	$Z$	0	$Z / 2 Z$
3	RP^3	Yes	$Z$	0	$Z / 2 Z$	$Z$
4	RP^4	No	$Z$	0	$Z / 2 Z$	0	$Z / 2 Z$
5	RP^5	Yes	$Z$	0	$Z / 2 Z$	0	$Z / 2 Z$	$Z$

@@ Line 12: / Line 12: @@
 <math>H_p(\mathbb{P}^n(\R)) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0\\ \Z/2\Z, & \qquad p \ \operatorname{even}, 0 < p \le n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
+==Cohomology groups with integer coefficients in tabular form==
+We illustrate how the cohomology groups work for small values of <math>n</math>. Note that for <math>p > n</math>, all cohomology groups <math>H^p</math> are zero, so we omit those cells for visual ease.
+{| class="sortable" border="1"
+! <math>n</math> !! Real projective space <math>\R\mathbb{P}^n</math> !! Orientable? !! <math>H_0</math> !! <math>H_1</math> !! <math>H_2</math> !! <math>H_3</math> !! <math>H_4</math> !! <math>H_5</math>
+|-
+| 1 || [[circle]] || Yes || <math>\mathbb{Z}</math> || <math>\mathbb{Z}</math>
+|-
+| 2 || [[real projective plane]] || No || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math>
+|-
+| 3 || [[RP^3]] || Yes || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math>  || <math>\mathbb{Z}</math>
+|-
+| 4 || [[RP^4]] || No || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math>
+|-
+| 5 || [[RP^5]] || Yes || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}</math>
+|}