Editing Real projective space (section)

===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}}