Real projective space

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

Basic properties[edit]

Construction[edit]

As with all projective spaces, RPn is formed by taking the quotient of Rn+1\{0} under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rn+1\{0} one can always find a λ such that λx has norm 1. There are precisely two such λ differing by sign.

Thus RPn can also be formed by identifying antipodal points of the unit n-sphere, Sn, in Rn+1.

One can further restrict to the upper hemisphere of Sn and merely identify antipodal points on the bounding equator. This shows that RPn is also equivalent to the closed n-dimensional disk, Dn, with antipodal points on the boundary, ∂Dn = Sn−1, identified.

Low-dimensional examples[edit]

RP1 is called the real projective line, which is topologically equivalent to a circle.

RP2 is called the real projective plane. This space cannot be embedded in R3. It can however be embedded in R4 and can be immersed in R3. The questions of embeddability and immersibility for projective n-space have been well-studied.[1]

RP3 is (diffeomorphic to) SO(3), hence admits a group structure; the covering map S3RP3 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[edit]

The antipodal map on the n-sphere (the map sending x to −x) generates a Z2 group action on Sn. As mentioned above, the orbit space for this action is RPn. This action is actually a covering space action giving Sn as a double cover of RPn. Since Sn is simply connected for n ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental group of RPn is Z2 when n > 1. (When n = 1 the fundamental group is Z due to the homeomorphism with S1). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Sn down to RPn.

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 Rp has sign (-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 RPn is orientable iff n+1 is even, i.e., n is odd.

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[edit]

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[edit]

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 open unit ball in Rn and the coordinate transition functions are smooth. This gives RPn a smooth structure.

CW structure[edit]

Real projective space RPn admits a CW structure 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−1RPn−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 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 Vk\Vk-1(lines in Vk but not Vk−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 RPn 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 Ui, g has nongenerate 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[edit]

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[edit]

Homotopy groups[edit]

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.

Explicitly, the fiber bundle is:

\mathbf{Z}_2 \to S^n \to \mathbf{RP}^n.

You might also write this as

S^0 \to S^n \to \mathbf{RP}^n

or

O(1) \to S^n \to \mathbf{RP}^n

by analogy with complex projective space.

The homotopy groups are:

\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}

Homology[edit]

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 : δDkRPk−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):

\mathrm{deg}(d_k) = 1 + (-1)^k.

Thus the integral homology is

H_i(\mathbf{RP}^n) =
\begin{cases}
\mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\
\mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \mbox{odd,}\\
0 & \mbox{else.}
\end{cases}

RPn is orientable iff n is odd, as the above homology calculation shows.

Infinite real projective space[edit]

The infinite real projective space is constructed as the direct limit or union of the finite projective spaces:

\mathbf{RP}^\infty := \lim_n \mathbf{RP}^n.

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

The double cover of this space is the infinite sphere S^\infty, which is contractible. The infinite projective space is therefore the Eilenberg-MacLane space K(Z2, 1).

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

Its cohomology ring modulo 2 is

H^*(\mathbf{RP}^\infty; \mathbf{Z}/2\mathbf{Z}) = \mathbf{Z}/2\mathbf{Z}[w_1],

where w_1 is the first Stiefel–Whitney class: it is the free \mathbf{Z}/2\mathbf{Z}-algebra on w_1, which has degree 1.

See also[edit]

Notes[edit]

  1. ^ See the table of Don Davis for a bibliography and list of results.

References[edit]