Editing Real projective space (section)

==Basic properties==
===Construction===
As with all [[projective space]]s, {{tmath|\mathbb{RP}^n}} is formed by taking the [[Quotient space (topology)|quotient]] of <math>\R^{n+1}\setminus \{0\}</math> under the [[equivalence relation]] {{tmath|x\sim \lambda x}} for all [[real number]]s {{tmath|\lambda\neq 0}}. For all {{tmath|x}} in <math>\R^{n+1}\setminus \{0\}</math> one can always find a {{tmath|\lambda}} such that {{tmath|\lambda x}} has [[Norm (mathematics)|norm]]&nbsp;1. There are precisely two such {{tmath|\lambda}} differing by sign. Thus {{tmath|\mathbb{RP}^n}} can also be formed by identifying [[antipodal point]]s of the unit {{tmath|n}}-[[sphere]], {{tmath|S^n}}, in <math>\R^{n+1}</math>.

One can further restrict to the upper hemisphere of {{tmath|S^n}} and merely identify antipodal points on the bounding equator. This shows that {{tmath|\mathbb{RP}^n}} is also equivalent to the closed {{tmath|n}}-dimensional disk, {{tmath|D^n}}, with antipodal points on the boundary, <math>\partial D^n=S^{n-1}</math>, identified.

===Low-dimensional examples===
* {{tmath|\mathbb{RP}^1}} is called the [[real projective line]], which is [[topology|topologically]] equivalent to a [[circle]]. Thinking of points of {{tmath|\mathbb{RP}^1}} as unit-norm complex numbers <math>z</math> up to sign, the diffeomorphism {{tmath|\mathbb{RP}^1 \to S^1}} is given by <math>z \mapsto z^2</math>. Geometrically, a line in <math>\mathbb{R}^2</math> is parameterized by an angle <math>\theta \in [0, \pi]</math> and the endpoints of this closed interval correspond to the same line.
* {{tmath|\mathbb{RP}^2}} is called the [[real projective plane]]. This space cannot be [[Embedding|embedded]] in {{tmath|\mathbb{R}^3}}. It can however be embedded in {{tmath|\mathbb{R}^4}} and can be [[Immersion (mathematics)|immersed]] in {{tmath|\mathbb{R}^3}} (see [[Boy's surface|here]]). The questions of embeddability and immersibility for projective {{tmath|n}}-space have been well-studied.<ref>See the table of Don Davis for a bibliography and list of results.</ref>
* {{tmath|\mathbb{RP}^3}} is [[diffeomorphic]] to [[SO(3)]], hence admits a group structure; the covering map {{tmath|S^3\to\mathbb{RP}^3}} is a map of groups Spin(3) → SO(3), where [[Spin group|Spin(3)]] is a [[Lie group]] that is the [[universal cover]] of SO(3).


===Topology===
The antipodal map on the {{tmath|n}}-sphere (the map sending {{tmath|x}} to {{tmath|-x}}) generates a [[cyclic group|'''Z'''<sub>2</sub>]] [[Group action (mathematics)|group action]] on {{tmath|S^n}}. As mentioned above, the orbit space for this action is {{tmath|\mathbb{RP}^n}}. This action is actually a [[covering space]] action giving {{tmath|S^n}} as a [[Double cover (topology)|double cover]] of {{tmath|\mathbb{RP}^n}}. Since {{tmath|S^n}} is [[simply connected]] for {{tmath|n\geq 2}}, it also serves as the [[universal cover]] in these cases. It follows that the [[fundamental group]] of {{tmath|\mathbb{RP}^n}} is {{tmath|\Z_2}} when {{tmath|n> 1}}. (When <math>n=1</math> the fundamental group is {{tmath|\Z}} due to the homeomorphism with {{tmath|S^1}}). A generator for the fundamental group is the closed [[curve]] obtained by projecting any curve connecting antipodal points in {{tmath|S^n}} down to {{tmath|\mathbb{RP}^n}}.

The projective {{tmath|n}}-space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its [[universal covering space]] is given by the antipody quotient map from the {{tmath|n}}-sphere, a [[simply connected]] space. It is a [[double covering group|double cover]]. The antipode map on {{tmath|\R^p}} has sign <math>(-1)^p</math>, so it is orientation-preserving if and only if {{tmath|p}} is even. The [[orientation character]] is thus: the non-trivial loop in <math>\pi_1(\mathbb{RP}^n)</math> acts as <math>(-1)^{n+1}</math> on orientation, so {{tmath|\mathbb{RP}^n}} is orientable if and only if {{tmath|n+1}} is even, i.e., {{tmath|n}} is odd.<ref>{{cite book|author1=J. T. Wloka|author2=B. Rowley |author3=B. Lawruk | title=Boundary Value Problems for Elliptic Systems|url=https://books.google.com/books?id=W7N8kyJB8NwC&pg=PA197| year=1995 | publisher=Cambridge University Press|isbn=978-0-521-43011-1|page=197}}</ref>

The projective {{tmath|n}}-space is in fact diffeomorphic to the submanifold of <math>\R^{(n+1)^2}</math> consisting of all symmetric {{tmath|(n+1)\times(n+1)}} matrices of [[Trace (linear algebra)|trace]] 1 that are also idempotent linear transformations.{{fact|date=April 2020}}