Editing Real projective space

{{short description|Type of topological space}}

In [[mathematics]], '''real projective space''', denoted {{tmath|\mathbb{RP}^n}} or {{tmath|\mathbb{P}_n(\R),}} is the [[topological space]] of [[Line (mathematics)|lines]] passing through the origin 0 in the [[real coordinate space|real space]] {{tmath|\R^{n+1}.}} It is a [[compact space|compact]], [[smooth manifold]] of [[dimension]] {{mvar|n}}, and is a special case {{tmath|\mathbf{Gr}(1, \R^{n+1})}} of a [[Grassmannian]] space.

==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}}

==Geometry of real projective spaces==
Real projective space admits a constant positive scalar curvature metric, coming from the double cover by the standard round sphere (the antipodal map is locally an isometry).

For the standard round metric, this has [[sectional curvature]] identically 1.

In the standard round metric, the measure of projective space is exactly half the measure of the sphere.
===Smooth structure===
Real projective spaces are [[smooth manifold]]s. On ''S<sup>n</sup>'', in homogeneous coordinates, (''x''<sub>1</sub>, ..., ''x''<sub>''n''+1</sub>), consider the subset ''U<sub>i</sub>'' with ''x<sub>i</sub>'' ≠ 0. Each ''U<sub>i</sub>'' is homeomorphic to the disjoint union of two open unit balls in '''R'''<sup>''n''</sup> that map to the same subset of '''RP'''<sup>''n''</sup> and the coordinate transition functions are smooth. This gives '''RP'''<sup>''n''</sup> a [[smooth structure]].

===Structure as a CW complex ===
Real projective space '''RP'''<sup>''n''</sup> admits the structure of a [[CW complex]] with 1 cell in every dimension. 

In homogeneous coordinates (''x''<sub>1</sub> ... ''x''<sub>''n''+1</sub>) on ''S<sup>n</sup>'', the coordinate neighborhood ''U''<sub>1</sub> = {(''x''<sub>1</sub> ... ''x''<sub>''n''+1</sub>) | ''x''<sub>1</sub> ≠ 0} can be identified with the interior of ''n''-disk  ''D<sup>n</sup>''. When ''x<sub>i</sub>'' = 0, one has '''RP'''<sup>''n''−1</sup>. Therefore the ''n''−1 skeleton of '''RP'''<sup>''n''</sup> is '''RP'''<sup>''n''−1</sup>, and the attaching map ''f'' : ''S''<sup>''n''−1</sup> → '''RP'''<sup>''n''−1</sup> is the 2-to-1 covering map. One can put
<math display="block">\mathbf{RP}^n = \mathbf{RP}^{n-1} \cup_f D^n.</math>

Induction shows that '''RP'''<sup>''n''</sup> is a CW complex with 1 cell in every dimension up to ''n''. 

The cells are [[Schubert cell]]s, as on the [[flag manifold]]. That is, take a complete [[flag (linear algebra)|flag]] (say the standard flag) 0 = ''V''<sub>0</sub> < ''V''<sub>1</sub> <...< ''V<sub>n</sub>''; then the closed ''k''-cell is lines that lie in ''V<sub>k</sub>''. Also the open ''k''-cell (the interior of the ''k''-cell) is lines in {{math|''V<sub>k</sub>'' \ ''V''<sub>''k''−1</sub>}} (lines in ''V<sub>k</sub>'' but not ''V''<sub>''k''−1</sub>). 

In homogeneous coordinates (with respect to the flag), the cells are
<math display="block">
\begin{array}{c}
[*:0:0:\dots:0] \\
{[}*:*:0:\dots:0] \\
\vdots \\
{[}*:*:*:\dots:*].
\end{array}</math>

This is not a regular CW structure, as the attaching maps are 2-to-1. However, its cover is a regular CW structure on the sphere, with 2 cells in every dimension; indeed, the minimal regular CW structure on the sphere.

In light of the smooth structure, the existence of a [[Morse function]] would show '''RP'''<sup>''n''</sup> is a CW complex. One such function is given by, in homogeneous coordinates,
<math display="block">g(x_1, \ldots, x_{n+1}) = \sum_{i=1} ^{n+1} i \cdot |x_i|^2.</math>

On each neighborhood ''U<sub>i</sub>'', ''g'' has nondegenerate critical point (0,...,1,...,0) where 1 occurs in the ''i''-th position with Morse index ''i''. This shows '''RP'''<sup>''n''</sup> is a CW complex with 1 cell in every dimension.

===Tautological bundles===
Real projective space has a natural [[line bundle]] over it, called the [[tautological bundle]]. More precisely, this is called the tautological subbundle, and there is also a dual ''n''-dimensional bundle called the tautological quotient bundle.

==Algebraic topology of real projective spaces==

===Homotopy groups===
The higher homotopy groups of '''RP'''<sup>''n''</sup> are exactly the higher homotopy groups of ''S<sup>n</sup>'', via the long exact sequence on homotopy associated to a [[fibration]].

Explicitly, the fiber bundle is: <math display="block">\mathbf{Z}_2 \to S^n \to \mathbf{RP}^n.</math>
You might also write this as <math display="block">S^0 \to S^n \to \mathbf{RP}^n</math>
or <math display="block">O(1) \to S^n \to \mathbf{RP}^n</math> by analogy with [[complex projective space]].

The homotopy groups are:
<math display="block">\pi_i (\mathbf{RP}^n) = \begin{cases}
0 & i = 0\\
\mathbf{Z}   & i = 1, n = 1\\
\mathbf{Z}/2\mathbf{Z} & i = 1, n > 1\\
\pi_i (S^n) & i > 1, n > 0.
\end{cases}</math>

===Homology===
The cellular chain complex associated to the above CW structure has 1 cell in each dimension 0, ..., ''n''. For each dimensional ''k'', the boundary maps ''d<sub>k</sub>'' : δ''D<sup>k</sup>'' → '''RP'''<sup>''k''−1</sup>/'''RP'''<sup>''k''−2</sup> is the map that collapses the equator on ''S''<sup>''k''−1</sup> and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2): 

<math display="block">\deg(d_k) = 1 + (-1)^k.</math>

Thus the integral [[cellular homology|homology]] is
<math display="block">H_i(\mathbf{RP}^n) = \begin{cases}
\mathbf{Z} & i = 0 \text{ or } i = n \text{ odd,}\\
\mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \text{odd,}\\
0 & \text{else.}
\end{cases}</math>

'''RP'''<sup>''n''</sup> is orientable if and only if ''n'' is odd, as the above homology calculation shows.

==Infinite real projective space==
The infinite real projective space is constructed as the [[direct limit]] or union of the finite projective spaces:
<math display="block">\mathbf{RP}^\infty := \lim_n \mathbf{RP}^n.</math>
This space is [[classifying space for O(n)|classifying space of ''O''(1)]], the first [[orthogonal group]].

The double cover of this space is the infinite sphere <math>S^\infty</math>, which is contractible. The infinite projective space is therefore the [[Eilenberg–MacLane space]] ''K''('''Z'''<sub>2</sub>, 1).  

For each nonnegative integer ''q'', the modulo 2 homology group <math>H_q(\mathbf{RP}^\infty; \mathbf{Z}/2) = \mathbf{Z}/2</math>.  

Its [[cohomology ring]] [[modulo (jargon)|modulo]] 2 is 
<math display="block">H^*(\mathbf{RP}^\infty; \mathbf{Z}/2\mathbf{Z}) = \mathbf{Z}/2\mathbf{Z}[w_1],</math>
where <math>w_1</math> is the first [[Stiefel–Whitney class]]: it is the free <math>\mathbf{Z}/2\mathbf{Z}</math>-algebra on <math>w_1</math>, which has degree 1.

Its [[cohomology ring]] with <math>\mathbf{Z}</math> coefficients is
<math display="block">H^*(\mathbf{RP}^{\infty};\mathbf{Z}) = \mathbf{Z}[\alpha]/(2\alpha), </math>
where <math>\alpha</math> has degree 2. It can be deduced from the [[chain map]] between cellular cochain complexes with <math>\mathbf{Z}</math> and <math>\mathbf{Z}/2</math> coefficients, which yield a ring homomorphism 
<math display="block">H^*(\mathbf{RP}^{\infty};\mathbf{Z}) \rightarrow H^*(\mathbf{RP}^{\infty};\mathbf{Z}/2\mathbf{Z})</math>
injective in positive dimensions, with image the even dimensional part of <math>H^*(\mathbf{RP}^{\infty};\mathbf{Z}/2\mathbf{Z})</math>. Alternatively, the result can also be obtained using the [[Universal coefficient theorem]].

==See also==
*[[Complex projective space]]
*[[Quaternionic projective space]]
*[[Lens space]]
*[[Real projective plane]]

==Notes==
<references/>

==References==
* [[Glen Bredon|Bredon, Glen]]. ''Topology and geometry'', Graduate Texts in Mathematics, Springer Verlag 1993, 1996
* {{cite web | last = Davis | first = Donald | title = Table of immersions and embeddings of real projective spaces | url = http://www.lehigh.edu/~dmd1/immtable | access-date = 22 Sep 2011}}
* {{cite book | last = Hatcher | first = Allen | author-link=Allen Hatcher| title = Algebraic Topology | publisher = [[Cambridge University Press]] | year = 2001 | isbn=978-0-521-79160-1 | url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html}}

{{DEFAULTSORT:Real Projective Space}}
[[Category:Algebraic topology]]
[[Category:Differential geometry]]
[[Category:Projective geometry]]