Real projective space

As with all projective spaces, RPⁿ is formed by taking the quotientofRⁿ⁺¹ ∖ {0} under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all xinRⁿ⁺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.

• 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 embeddedinR³. It can however be embedded in R⁴ and can be immersedinR³ (see here). The questions of embeddability and immersibility for projective n-space have been well-studied.^[1]
• RP³ is (diffeomorphicto) SO(3), hence admits a group structure; the covering map S³ → RP³ is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).

The antipodal map on the n-sphere (the map sending x to −x) generates a Z₂ group actiononSⁿ. As mentioned above, the orbit space for this action is RPⁿ. This action is actually a covering space action giving Sⁿ as a double coverofRPⁿ. Since Sⁿissimply connected for n ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental groupofRPⁿisZ₂ 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 R^p has sign ${\displaystyle (-1)^{p}}$ , so it is orientation-preserving if and only if p is even. The orientation character is thus: the non-trivial loop in ${\displaystyle \pi _{1}(\mathbf {RP} ^{n})}$  acts as ${\displaystyle (-1)^{n+1}}$  on orientation, so RPⁿ is orientable if and only if 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.^{[citation needed]}

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.

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 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 RPⁿasmooth structure.

Real projective space RPⁿ admits the structure of a CW complex 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 RPⁿ⁻¹. Therefore the n−1 skeleton of RPⁿisRPⁿ⁻¹, and the attaching map f : Sⁿ⁻¹ → RPⁿ⁻¹ is the 2-to-1 covering map. One can put $${\displaystyle \mathbf {RP} ^{n}=\mathbf {RP} ^{n-1}\cup _{f}D^{n}.}$$ 

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 <...< 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 $${\displaystyle {\begin{array}{c}[*:0:0:\dots :0]\\{[}*:*:0:\dots :0]\\\vdots \\{[}*:*:*:\dots :*].\end{array}}}$$ 

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, $${\displaystyle g(x_{1},\ldots ,x_{n+1})=\sum _{i=1}^{n+1}i\cdot |x_{i}|^{2}.}$$ 

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 RPⁿ is a CW complex with 1 cell in every dimension.

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.

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: $${\displaystyle \mathbf {Z} _{2}\to S^{n}\to \mathbf {RP} ^{n}.}$$  You might also write this as $${\displaystyle S^{0}\to S^{n}\to \mathbf {RP} ^{n}}$$ or$${\displaystyle O($$1)\to S^{n}\to \mathbf {RP} ^{n}}   by analogy with complex projective space.

The homotopy groups are: $${\displaystyle \pi _{i}(\mathbf {RP} ^{n})={\begin{cases}0&i=0\\\mathbf {Z} &i=1,n=1\\\mathbf {Z} /2\mathbf {Z} &i=1,n>1\\\pi _{i}(S^{n})&i>1,n>0.\end{cases}}}$$ 

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):

$${\displaystyle \deg(d_{k})=1+(-1)^{k}.}$$ 

Thus the integral homologyis$${\displaystyle H_{i}(\mathbf {RP} ^{n})={\begin{cases}\mathbf {Z} &i=0{\text{ or }}i=n{\text{ odd,}}\\\mathbf {Z} /2\mathbf {Z} &0<i<n,\ i\ {\text{odd,}}\\0&{\text{else.}}\end{cases}}}$$ 

RPⁿ is orientable if and only if n is odd, as the above homology calculation shows.

The infinite real projective space is constructed as the direct limit or union of the finite projective spaces: $${\displaystyle \mathbf {RP} ^{\infty }:=\lim _{n}\mathbf {RP} ^{n}.}$$  This space is classifying space of O(1), the first orthogonal group.

The double cover of this space is the infinite sphere ${\displaystyle S^{\infty }}$ , 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 ${\displaystyle H_{q}(\mathbf {RP} ^{\infty };\mathbf {Z} /2)=\mathbf {Z} /2}$ .

Its cohomology ring modulo 2 is $${\displaystyle H^{*}(\mathbf {RP} ^{\infty };\mathbf {Z} /2\mathbf {Z} )=\mathbf {Z} /2\mathbf {Z} [w_{1}],}$$  where ${\displaystyle w_{1}}$  is the first Stiefel–Whitney class: it is the free ${\displaystyle \mathbf {Z} /2\mathbf {Z} }$ -algebra on ${\displaystyle w_{1}}$ , which has degree 1.

• Complex projective space
• Quaternionic projective space
• Lens space
• Real projective plane

• 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.