# Real projective space

In mathematics, **real projective space**, or **RP**^{n} or , is the topological space of lines passing through the origin 0 in **R**^{n+1}. It is a compact, smooth manifold of dimension *n*, and is a special case **Gr**(1, **R**^{n+1}) of a Grassmannian space.

## Basic properties

### Construction

As with all projective spaces, **RP**^{n} is formed by taking the quotient of **R**^{n+1} \ {0} under the equivalence relation *x* ∼ *λx* for all real numbers λ ≠ 0. For all *x* in **R**^{n+1} \ {0} one can always find a *λ* such that *λx* 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

**RP**^{1} is called the real projective line, which is topologically equivalent to a circle.

**RP**^{2} is called the real projective plane. This space cannot be embedded in **R**^{3}. It can however be embedded in **R**^{4} and can be immersed in **R**^{3}. The questions of embeddability and immersibility for projective *n*-space have been well-studied.[1]

**RP**^{3} is (diffeomorphic to) SO(3), hence admits a group structure; the covering map *S*^{3} → **RP**^{3} is a map of groups Spin(3) → 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 **Z**_{2} 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 **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 closed curve obtained by projecting any curve connecting antipodal points in *S*^{n} down to **RP**^{n}.

The projective *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 *n*-sphere, a simply connected space. It is a double cover. The antipode map on **R**^{p} has sign , so it is orientation-preserving iff *p* is even. The orientation character is thus: the non-trivial loop in acts as on orientation, so **RP**^{n} is orientable iff *n* + 1 is even, i.e., *n* is odd.[2]

The projective *n*-space is in fact diffeomorphic to the submanifold of **R**^{(n+1)2} consisting of all symmetric (*n* + 1) × (*n* + 1) matrices of trace 1 that are also idempotent linear transformations.

## 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 manifolds. On *S ^{n}*, in homogeneous coordinates, (

*x*

_{1}...

*x*

_{n+1}), consider the subset

*U*with

_{i}*x*≠ 0. Each

_{i}*U*is homeomorphic to the open unit ball in

_{i}**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*. When

^{n}*x*= 0, one has

_{i}**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

Induction shows that **RP**^{n} is a CW complex with 1 cell in every dimension up to *n*.

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*. Also the open

_{k}*k*-cell (the interior of the

*k*-cell) is lines in

*V*\

_{k}*V*

_{k−1}(lines in

*V*but not

_{k}*V*

_{k−1}).

In homogeneous coordinates (with respect to the flag), the cells are

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,

On each neighborhood *U _{i}*,

*g*has nondegenerate 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.

### 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**^{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:

You might also write this as

or

by analogy with complex projective space.

The homotopy groups are:

### 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}* : δ

*D*→

^{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):

Thus the integral homology is

**RP**^{n} is orientable iff *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:

This space is classifying space of *O*(1), the first orthogonal group.

The double cover of this space is the infinite sphere , which is contractible. The infinite projective space is therefore the Eilenberg–MacLane space *K*(**Z**_{2}, 1).

For each nonnegative integer *q*, the modulo 2 homology group .

Its cohomology ring modulo 2 is

where is the first Stiefel–Whitney class: it is the free -algebra on , which has degree 1.

## Notes

- See the table of Don Davis for a bibliography and list of results.
- J. T. Wloka; B. Rowley; B. Lawruk (1995).
*Boundary Value Problems for Elliptic Systems*. Cambridge University Press. p. 197. ISBN 978-0-521-43011-1.

## References

- Bredon, Glen.
*Topology and geometry*, Graduate Texts in Mathematics, Springer Verlag 1993, 1996 - Davis, Donald. "Table of immersions and embeddings of real projective spaces". Retrieved 22 Sep 2011.
- Hatcher, Allen (2001).
*Algebraic Topology*. Cambridge University Press. ISBN 978-0-521-79160-1.