# Real projective space

In

mathematics ,**real projective space**, or**RP**^{"n"}is theprojective space of lines in**R**^{"n"+1}. It is a compact,smooth manifold of dimension "n", and a special case of aGrassmannian .**Construction**As with all projective spaces,

**RP**^{"n"}is formed by taking the quotient of**R**^{"n"+1}− {0} under theequivalence relation "x" ∼ λ"x" for allreal number s λ ≠ 0. For all "x" in**R**^{"n"+1}− {0} one can always find a λ such that λ"x" has norm 1. There are precisely two such λ differing by sign.Thus

**RP**^{"n"}can also be formed by identifyingantipodal point s of the unit "n"-sphere , "S"^{"n"}, in**R**^{"n"+1}.One can further restrict to the upper hemisphere of "S"

^{"n"}and merely identify antipodal points on the bounding equator. This shows that**RP**^{"n"}is also equivalent to the closed "n"-dimensional disk, "D"^{"n"}, with antipodal points on the boundary, ∂"D"^{"n"}= "S"^{"n"−1}, identified.**Low-dimensional examples**$mathbf\{RP\}^1$ is called the

real projective line , which is topologically equivalent to acircle .$mathbf\{RP\}^2$ is called the

real projective plane .$mathbf\{RP\}^3$ is (diffeomorphic to) "SO(3)", hence admits a group structure; the covering map $S^3\; o\; mathbf\{RP\}^3$ is a map of groups $operatorname\{Spin\}(3)\; o\; SO(3)$, where Spin(3) is a

Lie group that is theuniversal cover of SO(3).**Topology**The antipodal map on the "n"-sphere (the map sending "x" to −"x") generates a

**Z**_{2}group action on "S"^{"n"}. As mentioned above, the orbit space for this action is**RP**^{"n"}. This action is actually acovering space action giving "S"^{"n"}as adouble cover of**RP**^{"n"}. Since "S"^{"n"}issimply connected for "n" ≥ 2, it also serves as theuniversal cover in these cases. It follows that thefundamental group of**RP**^{"n"}is**Z**_{2}when "n > 1". (When "n = 1" the fundamental group is**Z**due to the homeomorphism with "S"^{"1"}). A generator for the fundamental group is the closedcurve obtained by projecting any curve connecting antipodal points in "S"^{"n"}down to**RP**^{"n"}.**Point-set topology**Some properties of projective n-space are:

* Projective 1-space is diffeomorphic to a circle

* Projective 2-space cannot be imbedded in

**R**^{3}. It can however be imbedded in**R**^{4}and can be immersed in**R**^{3}.* Projective n-space is in fact diffeomorphic to the submanifold of

**R**^{(n+1)2}consisting of all symmetric "(n+1)x(n+1)" matrices of trace 1 that are also idempotent linear transformations.* Projective n-space is compact connected and has a fundamental group isomorphic to the cyclic group of order 2 (the quotient map from the n-sphere to projective n-space is a double cover of projective n-space by a path connected space).

**Homotopy groups**The higher homotopy groups of $mathbf\{RP\}^n$ are exactly the higher homotopy groups of $S^n$, via the long exact sequence on homotopy associated to a

fibration .Explicitly, the fiber bundle is::$mathbf\{Z\}/2\; o\; S^n\; o\; mathbf\{RP\}^n.$You might also write this as:$S^0\; o\; S^n\; o\; mathbf\{RP\}^n$or:$O(1)\; o\; S^n\; o\; mathbf\{RP\}^n$by analogy with

complex projective space .The low homotopy groups are::$pi\_i\; mathbf\{RP\}^n\; =\; egin\{cases\}0\; i\; =\; 0\backslash mathbf\{Z\}/2\; i\; =\; 1\backslash 0\; 1\; i\; n\backslash mathbf\{Z\}\; i\; =\; nend\{cases\}$

**mooth structure**Real projective spaces are

smooth manifold s. On "S^{n}", in homogeneous coordinates, ("x"_{1}..."x"_{"n"+1}), consider the subset "U_{i}" with "x_{i}" ≠ 0. Each "U_{i}" is homeomorphic to the open unit ball in**R**^{"n"}and the coordinate transition functions are smooth. This gives**RP**^{"n"}a smooth structure.**CW structure**Real projective space

**RP**^{"n"}admits a CW structure with 1 cell in every dimension.In homogeneous coordinates ("x"

_{1}... "x"_{"n"+1}) on "S^{n}", the coordinate neighborhood "U_{1}" = {("x"_{1}... "x"_{"n"+1})|"x"_{1}≠ 0} can be identified with the interior of "n"-disk "D^{n}". When "x_{i}" = 0, one has**RP**^{"n" - 1}. Therefore the "n" - 1 skeleton of**RP**^{"n"}is**RP**^{"n" - 1}, and the attaching map "f": "S"^{"n"-1}→**RP**^{"n" - 1}is the 2-to-1 covering map. One can put:$mathbf\{RP\}^n\; =\; mathbf\{RP\}^\{n-1\}\; cup\_f\; D^n.$

Induction shows that

**RP**^{"n"}is a CW complex with 1 cell in every dimension.The cells are

Schubert cell s, as on theflag manifold .That is, take a complete flag (say the standard flag) 0 = "V"_{0}< "V"_{1}<...< "V_{n}"; then the closed "k"-cell is lines that lie in "V_{k}". Also the open "k"-cell (the interior of the "k"-cell) is lines in "V_{k}""V_{k-1}"(lines in "V_{k}" but not "V"_{"k" - 1}).In homogeneous coordinates (with respect to the flag), the cells are:$[*:0:0:dots:0]$:$[*:*:0:dots:0]$:$vdots$:$[*:*:*:dots:*]\; .$

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**^{"n"}is a CW complex. One such function is given by, in homogeneous coordinates,:$g(x\_1,\; cdots,\; x\_\{n+1\})\; =\; sum\_1\; ^\{n+1\}\; i\; cdot\; |x\_i|^2.$

On each neighborhood "U

_{i}", "g" has nongenerate critical point (0...,1,...0) where 1 occurs in the "i"-th position with Morse index "i". This shows**RP**^{"n"}is a CW complex with 1 cell in every dimension.**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

_{k}" : "δD^{k}" →**RP**^{"k"-1}/**RP**^{"k"-2}is the map that collapses the equator on "S"^{"k" - 1}and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2)::$mathrm\{deg\}(d\_k)\; =\; 1\; +\; (-1)^k.,$

Thus the integral homology is given by:

:$H\_i(mathbf\{RP\}^n)\; =\; egin\{cases\}mathbf\{Z\}\; i\; =\; 0\backslash mathbf\{Z/2\}\; 0,\; i\; mbox\{odd\}\backslash 0\; 0,\; mbox\{even\}\backslash mathbf\{z\}\; n\; mbox\{even\}.end\{cases\}\; math>$

**Orientability**$mathbf\{RP\}^n$ is orientable iff "n" is odd, as the above homology calculation shows. More concretely, the antipode map on $mathbf\{R\}^p$ hassign $(-1)^p$, so it is orientation-preserving iff "p" is even. The

orientation character is thus: the non-trivial loop in $pi\_1(mathbf\{RP\}^n)$ acts as $(-1)^\{n+1\}$ on orientation, so $mathbf\{RP\}^n$ is orientable iff "n+1" is even, i.e., "n" is odd.**Tautological bundles**Real projective space has a natural

line bundle over it, called thetautological bundle . More precisely, this is called the tautological subbundle, and there is also a dual "n"-dimensional bundle called the tautological quotient bundle.**Geometry**Real projective space admits a constant positive scalar curvature metric, coming from the double cover by the standard round sphere (the antipodal map is an isometry).

For the standard round metric, this has

sectional curvature identically 1.**Measure**In the standard round metric, the measure of projective space is exactly half the measure of the sphere.

**Infinite real projective space**Infinite real projective space is constructed asthe

direct limit or union of the finite projective spaces::$mathbf\{RP\}^infty\; :=\; lim\_n\; mathbf\{RP\}^n$Topologically, it is the Eilenberg-Mac Lane space $K(mathbf\{Z\}/2,1)$ (it is double-covered by the infinite sphere $S^infty$, which is contractible), and it is

BO(1) , theclassifying space forline bundle s.(Just as more generally, the infiniteGrassmannian is theclassifying space forvector bundle s.)Its cohomology ring at 2 is:$H^*(mathbf\{RP\}^infty;mathbf\{Z\}/2)\; =\; mathbf\{Z\}/2\; [w\_1]\; ,$where $w\_1$ is the first

Stiefel–Whitney class :it is the free $mathbf\{Z\}/2$-algebra on $w\_1$ (which has degree 1).**See also***

Complex projective space

*Quaternionic projective space

*Lens space

*Wikimedia Foundation.
2010.*

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