Cohomology of real projective space: Difference between revisions

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===Odd-dimensional projective space with coefficients in integers===
===Odd-dimensional projective space with coefficients in integers===


<math>H^p(\mathbb{P}^n(\R)) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0,n\\ \Z/2\Z, &\qquad p \ \operatorname{even}, 0 < p < n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
<math>H^p(\mathbb{P}^n(\R); \mathbb{Z}) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0,n\\ \Z/2\Z, &\qquad p \ \operatorname{even}, 0 < p < n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>


===Even-dimensional projective space with coefficients in integers===
===Even-dimensional projective space with coefficients in integers===


<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>
<math>H^p(\mathbb{P}^n(\R); \mathbb{Z}) = \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==
==Cohomology groups with integer coefficients in tabular form==
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| 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>
| 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>
|}
|}
==Facts used==
# [[uses::Homology of real projective space]]
# [[uses::Dual universal coefficients theorem]]
# [[uses::CW structure of real projective space]]
==Proof using homology groups==
===Case of odd dimension===
{{fillin}}
===Case of even dimension===
{{fillin}}
==Proof using cochain complex constructed from CW structure==
{{fillin}}

Revision as of 16:31, 27 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 cohomology group and the topological space/family is real projective space
Get more specific information about real projective space | Get more computations of cohomology group

Statement

Odd-dimensional projective space with coefficients in integers

Hp(Pn(R);Z)={Z,p=0,nZ/2Z,peven,0<p<n0,otherwise

Even-dimensional projective space with coefficients in integers

Hp(Pn(R);Z)={Z,p=0Z/2Z,peven,0<p≤n0,otherwise

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 Hp are zero, so we omit those cells for visual ease.

n Real projective space RPn Orientable? H0 H1 H2 H3 H4 H5
1 circle Yes Z Z
2 real projective plane No Z 0 Z/2Z
3 RP^3 Yes Z 0 Z/2Z Z
4 RP^4 No Z 0 Z/2Z 0 Z/2Z
5 RP^5 Yes Z 0 Z/2Z 0 Z/2Z Z

Facts used

  1. Homology of real projective space
  2. Dual universal coefficients theorem
  3. CW structure of real projective space

Proof using homology groups

Case of odd dimension

Fill this in later

Case of even dimension

Fill this in later

Proof using cochain complex constructed from CW structure

Fill this in later

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