Real projective space

In mathematics, real projective space, or RP^"n" is the projective space of lines in R^"n"+1. It is a compact, smooth manifold of dimension "n", and a special case of a Grassmannian.

Construction

As with all projective spaces, RP^"n" is formed by taking the quotient of R^"n"+1 − {0} under the equivalence relation "x" ∼ λ"x" for all real numbers 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 identifying antipodal points 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 a circle.

$mathbf\{RP\}^2$ 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 the universal cover of SO(3).

Topology

The antipodal map on the "n"-sphere (the map sending "x" to −"x") generates a Z₂ group action on "S"^"n". As mentioned above, the orbit space for this action is RP^"n". This action is actually a covering space action giving "S"^"n" as a double cover of RP^"n". Since "S"^"n" is simply connected for "n" ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental group of RP^"n" is Z₂ 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 closed curve 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 R³. It can however be imbedded in R⁴ and can be immersed in R³.

* 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::

mooth structure

Real projective spaces are smooth manifolds. On "Sⁿ", 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 "Sⁿ", the coordinate neighborhood "U₁" = {("x"₁ ... "x"_"n"+1)|"x"₁ ≠ 0} can be identified with the interior of "n"-disk "Dⁿ". 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 cells, as on the flag 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" &rarr; 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:

:

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 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.

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), the classifying space for line bundles.(Just as more generally, the infinite Grassmannian is the classifying space for vector bundles.)

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