Real projective space

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

Basic properties

Construction

As with all projective spaces, Template:Tmath is formed by taking the quotient of <math>\R^{n+1}\setminus \{0\}</math> under the equivalence relation Template:Tmath for all real numbers Template:Tmath. For all Template:Tmath in <math>\R^{n+1}\setminus \{0\}</math> one can always find a Template:Tmath such that Template:Tmath has norm 1. There are precisely two such Template:Tmath differing by sign. Thus Template:Tmath can also be formed by identifying antipodal points of the unit Template:Tmath-sphere, Template:Tmath, in <math>\R^{n+1}</math>.

One can further restrict to the upper hemisphere of Template:Tmath and merely identify antipodal points on the bounding equator. This shows that Template:Tmath is also equivalent to the closed Template:Tmath-dimensional disk, Template:Tmath, with antipodal points on the boundary, <math>\partial D^n=S^{n-1}</math>, identified.

Low-dimensional examples

• Template:Tmath is called the real projective line, which is topologically equivalent to a circle. Thinking of points of Template:Tmath as unit-norm complex numbers <math>z</math> up to sign, the diffeomorphism Template:Tmath is given by <math>z \mapsto z^2</math>. Geometrically, a line in <math>\mathbb{R}^2</math> is parameterized by an angle <math>\theta \in [0, \pi]</math> and the endpoints of this closed interval correspond to the same line.
• Template:Tmath is called the real projective plane. This space cannot be embedded in Template:Tmath. It can however be embedded in Template:Tmath and can be immersed in Template:Tmath (see here). The questions of embeddability and immersibility for projective Template:Tmath-space have been well-studied.<ref>See the table of Don Davis for a bibliography and list of results.</ref>
• Template:Tmath is diffeomorphic to SO(3), hence admits a group structure; the covering map Template:Tmath 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 Template:Tmath-sphere (the map sending Template:Tmath to Template:Tmath) generates a Z₂ group action on Template:Tmath. As mentioned above, the orbit space for this action is Template:Tmath. This action is actually a covering space action giving Template:Tmath as a double cover of Template:Tmath. Since Template:Tmath is simply connected for Template:Tmath, it also serves as the universal cover in these cases. It follows that the fundamental group of Template:Tmath is Template:Tmath when Template:Tmath. (When <math>n=1</math> the fundamental group is Template:Tmath due to the homeomorphism with Template:Tmath). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Template:Tmath down to Template:Tmath.

The projective Template:Tmath-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 Template:Tmath-sphere, a simply connected space. It is a double cover. The antipode map on Template:Tmath has sign <math>(-1)^p</math>, so it is orientation-preserving if and only if Template:Tmath is even. The orientation character is thus: the non-trivial loop in <math>\pi_1(\mathbb{RP}^n)</math> acts as <math>(-1)^{n+1}</math> on orientation, so Template:Tmath is orientable if and only if Template:Tmath is even, i.e., Template:Tmath is odd.<ref>Template:Cite book</ref>

The projective Template:Tmath-space is in fact diffeomorphic to the submanifold of <math>\R^{(n+1)^2}</math> consisting of all symmetric Template:Tmath matrices of trace 1 that are also idempotent linear transformations.Template:Fact

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, ..., 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 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 ... 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 RPⁿ is RPⁿ⁻¹, and the attaching map f : Sⁿ⁻¹ → RPⁿ⁻¹ is the 2-to-1 covering map. One can put <math display="block">\mathbf{RP}^n = \mathbf{RP}^{n-1} \cup_f D^n.</math>

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 Template:Math (lines in V_k but not V_k−1).

In homogeneous coordinates (with respect to the flag), the cells are <math display="block"> \begin{array}{c} [*:0:0:\dots:0] \\ {[}*:*:0:\dots:0] \\ \vdots \\ {[}*:*:*:\dots:*]. \end{array}</math>

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, <math display="block">g(x_1, \ldots, x_{n+1}) = \sum_{i=1} ^{n+1} i \cdot |x_i|^2.</math>

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.

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.

Explicitly, the fiber bundle is: <math display="block">\mathbf{Z}_2 \to S^n \to \mathbf{RP}^n.</math> You might also write this as <math display="block">S^0 \to S^n \to \mathbf{RP}^n</math> or <math display="block">O(1) \to S^n \to \mathbf{RP}^n</math> by analogy with complex projective space.

The homotopy groups are: <math display="block">\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}</math>

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):

<math display="block">\deg(d_k) = 1 + (-1)^k.</math>

Thus the integral homology is <math display="block">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}</math>

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 or union of the finite projective spaces: <math display="block">\mathbf{RP}^\infty := \lim_n \mathbf{RP}^n.</math> This space is classifying space of O(1), the first orthogonal group.

The double cover of this space is the infinite sphere <math>S^\infty</math>, 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 <math>H_q(\mathbf{RP}^\infty; \mathbf{Z}/2) = \mathbf{Z}/2</math>.

Its cohomology ring modulo 2 is <math display="block">H^*(\mathbf{RP}^\infty; \mathbf{Z}/2\mathbf{Z}) = \mathbf{Z}/2\mathbf{Z}[w_1],</math> where <math>w_1</math> is the first Stiefel–Whitney class: it is the free <math>\mathbf{Z}/2\mathbf{Z}</math>-algebra on <math>w_1</math>, which has degree 1.

Its cohomology ring with <math>\mathbf{Z}</math> coefficients is <math display="block">H^*(\mathbf{RP}^{\infty};\mathbf{Z}) = \mathbf{Z}[\alpha]/(2\alpha), </math> where <math>\alpha</math> has degree 2. It can be deduced from the chain map between cellular cochain complexes with <math>\mathbf{Z}</math> and <math>\mathbf{Z}/2</math> coefficients, which yield a ring homomorphism <math display="block">H^*(\mathbf{RP}^{\infty};\mathbf{Z}) \rightarrow H^*(\mathbf{RP}^{\infty};\mathbf{Z}/2\mathbf{Z})</math> injective in positive dimensions, with image the even dimensional part of <math>H^*(\mathbf{RP}^{\infty};\mathbf{Z}/2\mathbf{Z})</math>. 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

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References

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