In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.

Basic properties

Construction

Like all projective spaces, is formed by taking the quotient under the equivalence relation for all real numbers . For all one can always find a such that has norm1. There are precisely two such differing by sign. Thus, has the topology that is obtained by identifying antipodal points of the unit -sphere, , in .

One can alternatively restrict to the upper hemisphere of and merely identify antipodal points on the bounding equator. This shows that is also topologically equivalent to the closed -dimensional disk, , identified.

Low-dimensional examples

• is called the real projective line, which is topologically equivalent to a circle. Thinking of points of up to sign, the diffeomorphism and the endpoints of this closed interval correspond to the same line.

• is called the real projective plane. This space cannot be embedded in . It can however be embedded in and can be immersed in (see here). The questions of embeddability and immersibility for projective -space have been well-studied.

• is diffeomorphic to SO(3), hence admits a group structure; the covering map 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 -sphere (the map sending to ) generates a Z2 group action on . As mentioned above, the orbit space for this action is . This action is actually a covering space action giving as a double cover of . Since is simply connected for , it also serves as the universal cover in these cases. It follows that the fundamental group of is when the fundamental group is due to the homeomorphism with ). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in down to .

The projective -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 -sphere, a simply connected space. It is a double cover. The antipode map on , so it is orientation-preserving if and only if is even. The orientation character on orientation, so is orientable if and only if is even, i.e., is odd.

The projective consisting of all symmetric 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 Sn, in homogeneous coordinates, (x1, ..., xn+1), consider the subset Ui with xi ≠ 0. Each Ui is homeomorphic to the disjoint union of two open unit balls in Rn that map to the same subset of RPn and the coordinate transition functions are smooth. This gives RPn a smooth structure.

Structure as a CW complex

Real projective space RPn admits the structure of a CW complex with 1 cell in every dimension.

In homogeneous coordinates (x1 ... xn+1) on Sn, the coordinate neighborhood U1 = {(x1 ... xn+1) | x1 ≠ 0} can be identified with the interior of n-disk Dn. When xi = 0, one has RPn−1. Therefore the n−1 skeleton of RPn is RPn−1, and the attaching map f : Sn−1 → RPn−1

Induction shows that RPn 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 = V0 < V1 <...< Vn; then the closed k-cell is lines that lie in Vk. Also the open k-cell (the interior of the k-cell) is lines in (lines in Vk but not Vk−1).

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 RPn

On each neighborhood Ui, g has nondegenerate critical point (0,...,1,...,0) where 1 occurs in the i-th position with Morse index i. This shows RPn 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 RPn are exactly the higher homotopy groups of Sn, via the long exact sequence on homotopy associated to a fibration.

by analogy with complex projective space.

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 dk : δDk → RPk−1/RPk−2 is the map that collapses the equator on Sk−1 and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2):

Thus the integral homology

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

, which is contractible. The infinite projective space is therefore the Eilenberg–MacLane space K(Z2, 1).

For each nonnegative integer q.

Its cohomology ring modulo is the first Stiefel–Whitney class, which has degree 1.

Its cohomology ring has degree 2. It can be deduced from the chain map. 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

• Bredon, Glen. Topology and geometry, Graduate Texts in Mathematics, Springer Verlag 1993, 1996