Real projective space

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

1 Construction
2 Low-dimensional examples
3 Topology
4 Geometry
- 4.1 Measure
5 Infinite real projective space
6 See also

[edit] Construction

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.

[edit] 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 $S^3 \to \mathbf{RP}^3$ is a map of groups $\operatorname{Spin}(3) \to SO(3)$ , where Spin(3) is a Lie group that is the universal cover of SO(3).

[edit] Topology

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. A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Sⁿ down to RPⁿ.

[edit] Point-set topology

Real projective space is connected and compact, as it is quotient of a connected, compact space (the sphere).

[edit] 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:

$\mathbf{Z}/2 \to S^n \to \mathbf{RP}^n.$

You might also write this as

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

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

by analogy with complex projective space.

The low homotopy groups are:

$\pi_i \mathbf{RP}^n = \begin{cases} 0 & i = 0\\ \mathbf{Z}/2 & i = 1\\ 0 & 1 < i < n\\ \mathbf{Z} & i = n \end{cases}$

[edit] Smooth structure

Real projective spaces are smooth manifolds. On Sⁿ, in homogeneous coordinates, (x₁...x_n+1), consider the subset U_i with x_i ≠ 0. Each U_i is homeomorphic to the open unit ball in Rⁿ and the coordinate transition functions are smooth. This gives RPⁿ a smooth structure.

[edit] CW structure

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

In homogeneous coordinates (x₁ ... x_n+1) on Sⁿ, the coordinate neighborhood U₁ = {(x₁ ... x_n+1)|x₁ ≠ 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 RPⁿ 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 RPⁿ 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 = V₀ < V₁ <...< 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 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 U_i, 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.

[edit] 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:

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

[edit] Orientability

$\mathbf{RP}^n$ is orientable iff n is odd, as the above homology calculation shows. More concretely, the antipode map on $\mathbf{R}^p$ 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 $\mathbf{RP}^n$ is orientable iff n+1 is even, i.e., n is odd.

[edit] 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.

[edit] 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.

[edit] Measure

In the standard round metric, the measure of projective space is exactly half the measure of the sphere.

[edit] Infinite real projective space

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

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

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.)