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Real projective space

In mathematics, the real projective space of dimension n, denoted RPⁿ, is defined as the set of all one-dimensional subspaces (lines through the origin) of the vector space ℝⁿ⁺¹.^[1] Equivalently, it can be constructed as the quotient space of the n-sphere Sⁿ under the identification of antipodal points, where points x and -x are considered the same.^[2] This structure arises naturally in projective geometry as a completion of ℝⁿ by adding a "line at infinity," making parallel lines meet at points on this hyperplane.^[3] As a topological space, RPⁿ is a compact n-dimensional manifold that is Hausdorff, second-countable, and locally Euclidean, inheriting these properties from the sphere via the quotient map.^[2] It admits a smooth structure, allowing it to serve as a model for studying differential geometry and algebraic topology, though it is not embeddable in ℝⁿ for n ≥ 2 without self-intersections.^[2] For low dimensions, RP¹ is homeomorphic to the circle S¹, while RP² (the real projective plane) can be visualized as a disk with antipodal boundary points identified, exhibiting non-orientability.^[1]^[4] Real projective spaces play a central role in various fields, including the study of linear algebra over the reals, where points are represented by homogeneous coordinates [x₀: x₁: ⋯ : x_n], with equivalence under nonzero scalar multiplication.^[1] They are invariant under the action of the general linear group GL(n+1, ℝ), facilitating applications in computer vision, robotics, and the geometry of perspective projections.^[3] Furthermore, RPⁿ has a fundamental group isomorphic to ℤ/2ℤ for n ≥ 2, distinguishing it topologically from the sphere Sⁿ.^[2]

Definition and Construction

Quotient Space Construction

The real projective space \mathbb{RP}^n is defined as the set of all lines through the origin in \mathbb{R}^{n+1}, where each line is an unoriented one-dimensional subspace.^[5] Equivalently, \mathbb{RP}^n can be constructed as the quotient space S^n / \sim, where S^n \subset \mathbb{R}^{n+1} is the unit n-sphere and the equivalence relation \sim identifies each point x \in S^n with its antipode -x.^[5] In this construction, points in \mathbb{RP}^n are equivalence classes = \{x, -x\} for x \in S^n.^[5] The topology on \mathbb{RP}^n is the quotient topology induced by the canonical projection map \pi: S^n \to \mathbb{RP}^n defined by \pi(x) = .^[5] This means a subset U \subset \mathbb{RP}^n is open if and only if its preimage \pi^{-1}(U) is open in S^n.^[5] The quotient topology ensures that \mathbb{RP}^n inherits the relevant continuity properties from S^n, making \pi a continuous surjective map.^[5] Since the equivalence relation \sim is defined by a closed equivalence relation (the antipodal map is continuous and proper), the quotient space \mathbb{RP}^n is Hausdorff.^[5] The projection \pi is a local homeomorphism because, for any \in \mathbb{RP}^n, there exists a sufficiently small open neighborhood V of x in S^n such that V \cap (-V) = \emptyset, ensuring that \pi restricts to a homeomorphism from V to \pi(V).^[5] To see that \pi is a covering map, note that it is a two-sheeted map (each fiber has exactly two points), and for any open set W \subset \mathbb{RP}^n, the preimage \pi^{-1}(W) can be partitioned into evenly covered open sets in S^n that map homeomorphically onto W.^[5] Specifically, the antipodal action of \mathbb{Z}/2\mathbb{Z} on S^n is free and properly discontinuous, confirming the covering structure with S^n as the total space.^[5] Finally, \mathbb{RP}^n is compact for finite n because it is the continuous image of the compact space S^n under \pi.^[5] This quotient construction provides an abstract topological foundation, which can be coordinatized using homogeneous coordinates as an alternative representation.^[5]

Homogeneous Coordinates

Points in the real projective space \mathbb{RP}^n are represented using homogeneous coordinates [x_0 : x_1 : \dots : x_n], where (x_0, x_1, \dots, x_n) \in \mathbb{R}^{n+1} \setminus \{0\}, and two such tuples are equivalent if one is a nonzero scalar multiple of the other.^[6] This equivalence arises from the quotient space construction, where \mathbb{RP}^n consists of lines through the origin in \mathbb{R}^{n+1}.^[7] Each point in \mathbb{RP}^n thus corresponds to a one-dimensional subspace (line) of \mathbb{R}^{n+1}, excluding the origin.^[6] To coordinatize \mathbb{RP}^n, standard affine charts are defined as U_i = \{ \in \mathbb{RP}^n \mid x_i \neq 0 \} for i = 0, 1, \dots, n, where each U_i is homeomorphic to \mathbb{R}^n.^[8] On U_i, the coordinates are given by the affine map (x_0/x_i, x_1/x_i, \dots, \hat{x}_i/x_i, \dots, x_n/x_i), where the hat denotes omission of the i-th component.^[6] These n+1 charts cover \mathbb{RP}^n entirely, as every nonzero vector has at least one nonzero coordinate.^[7] The transition functions between overlapping charts U_i and U_j (where i \neq j) ensure a smooth gluing of these affine spaces.^[8] Specifically, these functions are rational and smooth on the overlaps; for example, transitioning from U_0 to U_1, if (y_1, y_2, \dots, y_n) are the coordinates on U_0 (with y_k = x_k / x_0), then on U_1 the coordinates are (z_0, z_2, \dots, z_n) where z_0 = 1 / y_1 and z_k = y_k / y_1 for k \geq 2. In general, the map involves inverting the affine coordinate corresponding to x_j / x_i and scaling the others accordingly, yielding diffeomorphisms.^[6] This demonstrates that \mathbb{RP}^n is a smooth manifold covered by n+1 copies of \mathbb{R}^n.^[7] Homogeneous coordinates also facilitate the description of projective hypersurfaces, which are subvarieties defined by the zero locus of a homogeneous polynomial f(x_0, \dots, x_n) = 0 of degree d.^[6] For instance, a linear equation a_0 x_0 + \dots + a_n x_n = 0 defines a projective hyperplane, corresponding to lines in \mathbb{R}^{n+1} orthogonal to the vector (a_0, \dots, a_n).^[7] Higher-degree equations similarly define hypersurfaces invariant under scalar multiplication, capturing geometric objects like conics or cubics in projective space.^[6]

Low-Dimensional Examples

The real projective line \mathbb{RP}^1 arises as the quotient space S^1 / \sim, where the unit circle S^1 has antipodal points identified, yielding a space homeomorphic to the circle S^1.^[2] This identification can be visualized by taking a semicircle and gluing its endpoints together, forming a loop that captures the topology of lines through the origin in \mathbb{R}^2.^[9] Geometrically, \mathbb{RP}^1 represents the projective line over the reals, where each point corresponds to a unique direction in the plane.^[6] In homogeneous coordinates, points of \mathbb{RP}^1 are equivalence classes [x : y] with (x, y) \in \mathbb{R}^2 \setminus \{(0,0)\}, where [x : y] = [\lambda x : \lambda y] for \lambda \in \mathbb{R} \setminus \{0\}.^[6] The affine chart y \neq 0 normalizes to y = 1, parametrizing points as [x : 1] for x \in \mathbb{R}, which embeds the real line; the remaining point [1 : 0] serves as the point at infinity, compactifying \mathbb{R} into a circle.^[9] The real projective plane \mathbb{RP}^2 is constructed as the quotient of the 2-sphere S^2 by antipodal identification, or equivalently, as the unit disk D^2 with opposite boundary points glued together (i.e., x \sim -x for x \in \partial D^2 = S^1).^[2] This disk model highlights the non-orientable topology, where the boundary identifications create a closed surface without embedding in \mathbb{R}^3.^[4] Another visualization is the hemisphere S^2_+ with antipodal points on the equatorial circle identified, representing lines through the origin in \mathbb{R}^3.^[6] In homogeneous coordinates, points of \mathbb{RP}^2 are [x : y : z] with (x, y, z) \in \mathbb{R}^3 \setminus \{(0,0,0)\}, up to nonzero scalar multiples, each corresponding to a unique line in \mathbb{R}^3.^[4] This algebraic description aligns with the geometric models, where the projective plane compactifies \mathbb{R}^2 by adding a line at infinity.^[6]

Topological Properties

Fundamental Group and Covering Space

The real projective space \mathbb{RP}^n admits a double covering map p: S^n \to \mathbb{RP}^n given by identifying antipodal points on the n-sphere S^n, where S^n is simply connected for n \geq 2.^[5] This covering space is universal because S^n has trivial fundamental group in these dimensions, and the deck transformation group consists of the identity map and the antipodal map a: S^n \to S^n defined by a(x) = -x, which generates the cyclic group \mathbb{Z}_2.^[5] To compute the fundamental group \pi_1(\mathbb{RP}^n) for n \geq 2, consider a based loop \gamma in \mathbb{RP}^n at the basepoint corresponding to the line through (1, 0, \dots, 0). By the path-lifting property of covering spaces, \gamma lifts to a unique path \tilde{\gamma} in S^n starting at (1, 0, \dots, 0) and ending at either the same point or its antipode (-1, 0, \dots, 0).^[5] If the endpoint is the starting point, then \gamma is nullhomotopic in \mathbb{RP}^n, as it lifts to a loop in the simply connected S^n. Otherwise, the endpoint is the antipode, and the loop $2\gamma (traversed twice) lifts to a closed loop in S^n, which is nullhomotopic; thus, $2[\gamma] = 0 in \pi_1(\mathbb{RP}^n). Moreover, no nontrivial loop lifts to a closed loop, so \pi_1(\mathbb{RP}^n) \cong \mathbb{Z}_2.^[5] The homotopy-lifting property ensures that homotopies in \mathbb{RP}^n lift to homotopies in S^n, preserving the algebraic structure of the fundamental group under the covering map.^[5] This isomorphism \pi_1(\mathbb{RP}^n) \cong \mathbb{Z}_2 for n \geq 2 reflects the nontrivial connectivity introduced by the antipodal identification. For n=1, \mathbb{RP}^1 is homeomorphic to the circle S^1, as the quotient of S^1 by antipodes identifies opposite points on the circle, yielding another circle.^[5] Consequently, \pi_1(\mathbb{RP}^1) \cong \mathbb{Z}, the infinite cyclic group generated by loops winding around the circle.^[5]

Orientability and Manifold Structure

The real projective space \mathbb{RP}^n admits a smooth atlas constructed from affine charts based on homogeneous coordinates [x_0 : \dots : x_n], where points represent lines through the origin in \mathbb{R}^{n+1}. The space is covered by the open sets U_i = \{ [x_0 : \dots : x_n] \in \mathbb{RP}^n \mid x_i \neq 0 \} for i = 0, \dots, n. The chart \phi_i: U_i \to \mathbb{R}^n is defined by

\phi_i([x_0 : \dots : x_n]) = \left( \frac{x_0}{x_i}, \dots, \frac{x_{i-1}}{x_i}, \frac{x_{i+1}}{x_i}, \dots, \frac{x_n}{x_i} \right),

with the inverse \phi_i^{-1}(y_1, \dots, y_n) = [y_1 : \dots : y_{i-1} : 1 : y_i : \dots : y_n], inserting 1 in the i-th position. These charts provide local trivializations, identifying neighborhoods in \mathbb{RP}^n with open subsets of \mathbb{R}^n.^[10] The transition maps between overlapping charts are restrictions of linear fractional transformations and hence smooth. For instance, on \phi_0(U_0 \cap U_1), the map \phi_1 \circ \phi_0^{-1}: (y_1, \dots, y_n) \mapsto \left( \frac{1}{y_1}, \frac{y_2}{y_1}, \dots, \frac{y_n}{y_1} \right) is a diffeomorphism wherever y_1 \neq 0. In general, for i \neq j, \phi_j \circ \phi_i^{-1} takes the form of rational functions with non-vanishing denominators on the domain, ensuring C^\infty compatibility across the atlas. This defines a smooth structure on \mathbb{RP}^n, inheriting the differential structure from \mathbb{R}^{n+1} \setminus \{0\} via the quotient by nonzero scalar multiplication, a free and proper smooth action.^[10]^[11] With this atlas, \mathbb{RP}^n is a smooth manifold of dimension n. It is Hausdorff and second countable as a quotient of the second countable space \mathbb{R}^{n+1} \setminus \{0\} by an open equivalence relation. Compactness follows from the continuous quotient map S^n \to \mathbb{RP}^n by the antipodal identification, since S^n is compact. As an image under charts to open sets in \mathbb{R}^n, \mathbb{RP}^n is a closed manifold without boundary.^[10]^[11] The manifold \mathbb{RP}^n is orientable if and only if n is odd. For even n, non-orientability is witnessed by the nontrivial first Stiefel-Whitney class w_1(T\mathbb{RP}^n) \neq 0. The total Stiefel-Whitney class is w(T\mathbb{RP}^n) = (1 + a)^{n+1} \in H^*(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}), where a generates H^1(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}; thus, w_1 = (n+1)a, which is a \neq 0 when n is even (since n+1 is odd). For odd n, n+1 is even, so w_1 = 0, implying orientability.^[11]^[12] This aligns with the double covering p: S^n \to \mathbb{RP}^n by the antipodal map, whose differential at each point is multiplication by -1 on the tangent space, with determinant (-1)^{n+1}. When n is odd, this determinant is +1, so the action preserves orientation and the quotient is orientable; when n is even, the determinant is -1, reversing orientation and yielding a non-orientable quotient.^[12]

CW Complex Structure

The real projective space \mathbb{RP}^n admits a CW complex structure consisting of exactly one open cell e^k in each dimension k from 0 to n. This decomposition arises inductively by viewing \mathbb{RP}^n as the quotient of the n-sphere S^n under the antipodal identification, where the cells correspond to the images of suitable hemispherical regions under this quotient map.^[5]^[13] The 0-cell e^0 is a single point, representing \mathbb{RP}^0. For k \geq 1, the k-cell e^k is the quotient of the open k-disk D^k by identifying antipodal points on its boundary S^{k-1}, effectively \mathbb{RP}^k minus its (k-1)-skeleton. This cell is attached to the (k-1)-skeleton along the boundary via the quotient map S^{k-1} \to \mathbb{RP}^{k-1}, which identifies antipodal points and thus realizes a degree-2 covering map (double cover). For the specific case of the 1-cell, the attaching map S^0 \to \mathbb{RP}^0 sends both points of the 0-sphere to the single point of \mathbb{RP}^0, consistent with the antipodal identification on S^0. For higher dimensions, the attaching maps are the standard antipodal quotients S^k \to \mathbb{RP}^k. This structure ensures that the CW complex is regular, as the attaching maps are continuous and the cells are openly embedded.^[5]^[13] The k-skeleton of this CW complex is precisely \mathbb{RP}^k, built by including all cells up to dimension k. This inductive relation highlights how the full space \mathbb{RP}^n emerges by successively adjoining higher-dimensional cells to lower projective subspaces.^[5]^[13] The Euler characteristic of \mathbb{RP}^n in this CW structure is the alternating sum of the number of cells: \chi(\mathbb{RP}^n) = \sum_{k=0}^n (-1)^k. This evaluates to 1 when n is even and 0 when n is odd, reflecting the parity-dependent topological properties of the space.^[5]

Geometric Structures

Tautological Line Bundle

The tautological line bundle over the real projective space \mathbb{RP}^n, denoted \tau: E \to \mathbb{RP}^n, is the canonical rank-1 real vector bundle whose fiber over each point [\ell] \in \mathbb{RP}^n, representing a 1-dimensional subspace \ell \subset \mathbb{R}^{n+1}, is the line \ell itself.^[14] The total space E is the subset \{ ([\ell], v) \mid v \in \ell \} \subset \mathbb{RP}^n \times \mathbb{R}^{n+1}, equipped with the subspace topology and vector space structure on each fiber.^[14] Equivalently, the total space E (including the zero section) arises as the associated bundle to the principal \mathbb{R}^*-bundle \mathbb{R}^{n+1} \setminus \{0\} \to \mathbb{RP}^n, but the pair construction directly embeds it as a subbundle of the trivial rank-(n+1) bundle over \mathbb{RP}^n. To describe its bundle structure, consider the standard affine charts U_i = \{ \in \mathbb{RP}^n \mid x_i \neq 0 \} for i = 0, \dots, n, where

denotes [homogeneous coordinates](/page/Homogeneous_coordinates) in $\mathbb{R}^{n+1} \setminus \{0\}$. Over each $U_i$, the bundle restricts to a trivial [line bundle](/page/Line_bundle) via the local trivialization $\phi_i: \pi^{-1}(U_i) \to U_i \times \mathbb{R}$ defined by sending $(, v)$ to $(, t)$, where $t$ is the coordinate such that $v = t \cdot \tilde{x}$ and $\tilde{x}$ is the representative of

normalized so that its i-th component is 1 (i.e., \tilde{x}_i = 1, \tilde{x}_j = x_j / x_i for j \neq i).^[14] On overlaps U_i \cap U_j, the transition function g_{ij}: U_i \cap U_j \to \mathbb{R}^* is given by g_{ij}() = x_j / x_i, which reflects the change in normalization between charts and ensures the vector space structure is consistently defined. These transition maps, often called inversion maps due to their reciprocal form g_{ji} = 1 / g_{ij}, confirm that \tau is a genuine vector bundle rather than merely a fiber bundle.^[14] The tautological line bundle \tau is a non-trivial rank-1 real vector bundle over \mathbb{RP}^n for n \geq 1. Its non-triviality follows from the absence of a global non-zero section: any continuous section s: \mathbb{RP}^n \to E would assign to each line \ell a non-zero vector in \ell, yielding a continuous nowhere-zero map from \mathbb{RP}^n to S^n (via normalization), which is impossible by the Borsuk–Ulam theorem or the non-vanishing of the first Stiefel–Whitney class w_1(\tau) \neq 0.^[14] The dual bundle \tau^* is the hyperplane bundle, whose fiber over [\ell] consists of linear functionals on \mathbb{R}^{n+1} modulo those vanishing on \ell, and the transition functions for \tau^* are the inverses g_{ij}^{-1} = x_i / x_j. As real line bundles are isomorphic to their duals (since w_1(\tau^*) = w_1(\tau)), \tau \cong \tau^* over \mathbb{RP}^n, making the hyperplane bundle the canonical realization of this dual.^[14]

Riemannian Metric and Geometry

The real projective space \mathbb{RP}^n can be endowed with a standard Riemannian metric induced from the round metric on its double covering space, the n-sphere S^n. The round metric on S^n \subset \mathbb{R}^{n+1} is the one with constant radius 1, which is invariant under the action of the orthogonal group O(n+1). Since the antipodal identification map x \mapsto -x on S^n is an isometry with respect to this metric, it descends to a well-defined Riemannian metric on the quotient \mathbb{RP}^n = S^n / \{\pm 1\}, preserving the O(n+1)-invariance.^[15] This construction equips \mathbb{RP}^n with a smooth Riemannian structure where the projection \pi: S^n \to \mathbb{RP}^n is a local isometry.^[16] The geodesics on \mathbb{RP}^n arise as the images under \pi of the great circles on S^n. A great circle on S^n (of length $2\pi) projects to a closed geodesic on \mathbb{RP}^n of length \pi, as the identification equates points separated by an angle of \pi.^[17] These projected geodesics are minimizing up to length \pi/2, which is the injectivity radius of \mathbb{RP}^n; beyond this distance, the exponential map ceases to be injective due to the antipodal identification reaching the cut locus.^[18] As a Riemannian quotient by an isometry, \mathbb{RP}^n inherits constant sectional curvature 1 from S^n. This positive constant curvature characterizes \mathbb{RP}^n as a space form, with all two-planes in tangent spaces exhibiting the same curvature value.^[15] Note that alternative normalizations exist, such as scaling the metric on S^n to radius $1/\sqrt{2} (yielding curvature 2 on \mathbb{RP}^n) or radius $1/2 (yielding curvature 4), but the standard choice aligns with the unit sphere for consistency with spherical geometry.^[19] The totally geodesic submanifolds of \mathbb{RP}^n are precisely its projective subspaces \mathbb{RP}^k for $0 \leq k < n. These arise as quotients of great k-spheres in S^n and inherit the constant sectional curvature 1, embedding isometrically into the ambient metric.^[15]

Projective Transformations

The group of projective transformations of \mathbb{RP}^n is the projective general linear group \mathrm{PGL}(n+1, \mathbb{R}), which is isomorphic to the automorphism group \mathrm{Aut}(\mathbb{RP}^n). This group is defined as the quotient \mathrm{GL}(n+1, \mathbb{R}) / \mathbb{R}^*, where \mathbb{R}^* denotes the multiplicative group of nonzero real numbers, and it acts on \mathbb{RP}^n via linear transformations on \mathbb{R}^{n+1} modulo scalar multiples.^[20] The action preserves the structure of lines through the origin, thereby maintaining the projective geometry of incidence relations between points, lines, and hyperplanes.^[20] The action of \mathrm{PGL}(n+1, \mathbb{R}) on points of \mathbb{RP}^n is given explicitly by \mapsto [A x], where A \in \mathrm{GL}(n+1, \mathbb{R}) and $$ denotes the equivalence class of x \in \mathbb{R}^{n+1} \setminus \{0\} under scalar multiplication. This induces a transitive action on \mathbb{RP}^n, meaning any two points can be mapped to each other by some element of the group, with orbits corresponding to projective subspaces or more general projective varieties defined by the invariant sets under subgroups.^[20] Fixed points of a projective transformation correspond to the projective eigenspaces of the representing linear map A (up to scalar multiple), where an eigenspace \mathrm{Eig}_\lambda(A) projects to a fixed projective subspace; for instance, in \mathbb{RP}^2, a diagonalizable transformation may fix a point and a line as invariant projective subspaces.^[20] Projective transformations admit classification based on the properties of the underlying linear maps up to scaling, such as rank, nullity, and diagonalizability over \mathbb{R}. Nonsingular transformations have full rank n+1, while singular ones possess a "bad set" of fixed or undefined behavior; over \mathbb{R}, diagonalizable cases occur when all eigenvalues are real, leading to decompositions into invariant eigenspaces that project to fixed projective components.^[20] In the low-dimensional case of \mathbb{RP}^1, elements of \mathrm{PGL}(2, \mathbb{R}) are classified by the number of fixed points (zero, one, or two), and the cross-ratio of four distinct points, defined as [(x,y;w,z) = \frac{(w-x)/(w-z)}{(y-x)/(y-z)}], remains invariant under these transformations, serving as a fundamental projective invariant that distinguishes quadruples up to equivalence.^[21]

Algebraic Topology

Homology Groups

The real projective space \mathbb{RP}^n admits a CW complex structure with exactly one cell in each dimension from 0 to n, which facilitates the computation of its singular homology groups via the cellular chain complex.^[5] The cellular chain groups are C_k(\mathbb{RP}^n; \mathbb{Z}) \cong \mathbb{Z} for $0 \leq k \leq n and 0 otherwise, generated by the fundamental class of the unique k-cell.^[5] The boundary maps in this complex are determined by the degrees of the attaching maps, which arise from the quotient construction \mathbb{RP}^n = S^n / \sim, where \sim identifies antipodal points.^[22] With \mathbb{Z}_2 coefficients, the boundary maps vanish because the attaching maps have even degree, making the chain complex acyclic except for the generators.^[5] Thus, the mod-2 homology is H_k(\mathbb{RP}^n; \mathbb{Z}_2) \cong \mathbb{Z}_2 for each $0 \leq k \leq n, generated by the fundamental classes of the cells, and H_k(\mathbb{RP}^n; \mathbb{Z}_2) = 0 for k > n.^[5]^[22] For integer coefficients, the boundary maps alternate: \partial_k = 0 when k is odd, and \partial_k(e_k) = 2 e_{k-1} when k is even, reflecting the degree-2 antipodal map on the attaching spheres.^[5] This structure induces \mathbb{Z}_2-torsion in odd dimensions and periodicity in the homology, yielding:

H_k(\mathbb{RP}^n; \mathbb{Z}) \cong \begin{cases} \mathbb{Z} & k = 0, \\ \mathbb{Z}_2 & 0 < k < n,\ k\ \text{odd}, \\ 0 & 0 < k \leq n,\ k\ \text{even}, \\ \mathbb{Z} & k = n\ \text{odd}, \\ 0 & k = n\ \text{even}, \end{cases}

with H_k(\mathbb{RP}^n; \mathbb{Z}) = 0 for k > n.^[5]^[22] In particular, H_1(\mathbb{RP}^n; \mathbb{Z}) \cong \mathbb{Z}_2 for n \geq 1, which follows from the long exact sequence of the double covering S^n \to \mathbb{RP}^n, where the action of the deck transformation induces multiplication by 2 on H_1(S^n; \mathbb{Z}), producing the torsion.^[5]

Cohomology Ring

The mod 2 cohomology ring of the real projective space \mathbb{RP}^n is given by

H^*(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} / (x^{n+1}),

where x \in H^1(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) is the generator.^[23]^[24] This polynomial ring structure arises from the cellular cochain complex of \mathbb{RP}^n, where the nonzero cup products of the fundamental coclasses in dimensions 0 through n yield powers of x up to degree n.^[24] Alternatively, the structure follows from the projective bundle theorem applied to the tautological line bundle \gamma_1^n \to \mathbb{RP}^n, which identifies \mathbb{RP}^n as the projectivization P(\underline{\mathbb{R}}^{n+1}) and induces the multiplicative formula via the associated Thom isomorphism.^[23] The generator x coincides with the first Stiefel-Whitney class w_1(\gamma_1^n) of the tautological line bundle \gamma_1^n.^[25] This identification follows from the axiomatic definition of Stiefel-Whitney classes, where w_1(\gamma_1^\infty) generates H^*(\mathbb{RP}^\infty; \mathbb{Z}/2\mathbb{Z}) as the mod 2 reduction of the orientation class, and the finite-dimensional case truncates accordingly.^[25]^[26] The Steenrod squares act nontrivially on this ring, providing additional structure beyond the cup product. Specifically, for the generator powers, \mathrm{Sq}^i(x^j) = \binom{j}{i} x^{j+i} \pmod{2}, so in particular \mathrm{Sq}^1(x^k) = k x^{k+1} \pmod{2}.^[27] These operations are nontrivial up to degree n, as the binomial coefficients modulo 2 (governed by Lucas' theorem) ensure that higher squares connect basis elements within the truncated polynomial ring, reflecting the unstable cohomology operations on the cells of \mathbb{RP}^n.^[27] The nonzero class x relates directly to the orientability of \mathbb{RP}^n. The first Stiefel-Whitney class of the tangent bundle is w_1(T\mathbb{RP}^n) = (n+1) x \pmod{2}, so when n is even (making n+1 odd), w_1(T\mathbb{RP}^n) = x \neq 0, implying \mathbb{RP}^n is non-orientable.^[26] This computation uses the splitting T\mathbb{RP}^n \oplus \underline{\mathbb{R}} \cong (n+1) \gamma_1^n and the Whitney sum formula for Stiefel-Whitney classes.^[26]

Homotopy Groups

The homotopy groups of the real projective space \mathbb{RP}^n are closely related to those of the n-sphere S^n via the double covering map p: S^n \to \mathbb{RP}^n, which identifies antipodal points.^[28] For n=1, \mathbb{RP}^1 is homeomorphic to S^1, so \pi_1(\mathbb{RP}^1) \cong \mathbb{Z} and \pi_k(\mathbb{RP}^1) = 0 for k \geq 2.^[29] For n \geq 2, the sphere S^n is simply connected, making this covering universal with deck transformation group \mathbb{Z}/2\mathbb{Z}. The long exact sequence in homotopy groups for the associated fibration \mathbb{Z}/2 \to S^n \to \mathbb{RP}^n yields \pi_1(\mathbb{RP}^n) \cong \mathbb{Z}/2\mathbb{Z}, since \pi_k(\mathbb{Z}/2) = 0 for k \geq 1 and \pi_0(\mathbb{Z}/2) \cong \mathbb{Z}/2. For k \geq 2, the same sequence implies \pi_k(\mathbb{RP}^n) \cong \pi_k(S^n), as the fiber contributes trivially in positive dimensions.^[30]^[31] In particular, \pi_2(\mathbb{RP}^2) \cong \mathbb{Z}. More generally, the isomorphism holds for all k \geq 2, though the \mathbb{Z}/2-action on S^n via the antipodal map can influence representatives in low dimensions relative to n; for instance, \pi_2(\mathbb{RP}^n) = 0 for n > 2. The groups \pi_k(S^n) are zero for $1 < k < n and nontrivial in higher unstable ranges, computed via methods like the Freudenthal suspension theorem, which establishes isomorphisms \pi_k(S^n) \to \pi_{k+1}(S^{n+1}) for k < 2n-1.^[30] To compute these recursively across dimensions, the cofiber sequence \mathbb{RP}^{n-1} \to \mathbb{RP}^n \to S^n provides a long exact sequence \cdots \to \pi_k(\mathbb{RP}^{n-1}) \to \pi_k(\mathbb{RP}^n) \to \pi_k(S^n) \to \pi_{k-1}(\mathbb{RP}^{n-1}) \to \cdots. In the range k < 2n-1, Freudenthal suspension ensures stability, confirming the isomorphism to \pi_k(S^n) for k \geq n+1 and aligning with the covering space result overall, with the \mathbb{Z}/2-action accounting for any low-dimensional adjustments.^[30]^[32] Unlike complex or quaternionic projective spaces, the real case lacks Bott periodicity in its finite-dimensional homotopy groups, though the stable limit as n \to \infty relates to the homotopy of the orthogonal group O(n) via the classifying space construction for orthogonal bundles.^[30]

Infinite Real Projective Space

Construction and Properties

The infinite real projective space \RP^\infty is constructed as the direct limit \colim_{n \to \infty} \RP^n of the finite-dimensional real projective spaces, where the inclusion maps \RP^n \hookrightarrow \RP^{n+1} are given by [x_0 : \cdots : x_n] \mapsto [x_0 : \cdots : x_n : 0].^[5] This direct limit equips \RP^\infty with the topology of an infinite CW complex, consisting of one open cell e^k in each dimension k \geq 0, extending the CW structure of the finite \RP^n.^[5] The finite-dimensional subspaces \RP^n are densely embedded in \RP^\infty, as they form the finite skeletons of this CW complex.^[5] A key property of \RP^\infty is its universal cover, which is the infinite-dimensional sphere S^\infty, obtained as the direct limit of the Stiefel manifolds V_{n+1,1} \cong S^n and is contractible.^[5] This covering map S^\infty \to \RP^\infty is the infinite analogue of the antipodal quotient S^n \to \RP^n. The space \RP^\infty itself is non-compact, reflecting its infinite dimensionality as an increasing union of compact finite approximations.^[5] Mapping spaces into \RP^\infty, such as [\cdot, \RP^\infty], are typically endowed with the compact-open topology to ensure continuity in the direct limit construction.^[5] The homotopy type of \RP^\infty is that of an Eilenberg-MacLane space K(\mathbb{Z}/2\mathbb{Z}, 1), with fundamental group \pi_1(\RP^\infty) \cong \mathbb{Z}/2\mathbb{Z} and all higher homotopy groups vanishing: \pi_k(\RP^\infty) = 0 for k \geq 2.^[5] This follows directly from the contractibility of the universal cover S^\infty, implying that \RP^\infty is the classifying space for principal \mathbb{Z}/2\mathbb{Z}-bundles up to homotopy.^[5]

Classifying Space for O(n)

The infinite real projective space \mathbb{RP}^\infty serves as the classifying space BO(1) for the orthogonal group O(1), which is isomorphic to \mathbb{Z}/2\mathbb{Z}.^[5] Equivalently, \mathbb{RP}^\infty is the Eilenberg-MacLane space K(\mathbb{Z}_2, 1) = B\mathbb{Z}_2, characterized by its fundamental group \pi_1(\mathbb{RP}^\infty) \cong \mathbb{Z}_2 and trivial higher homotopy groups \pi_i(\mathbb{RP}^\infty) = 0 for i \geq 2.^[5] This structure implies that principal \mathbb{Z}_2-bundles, or equivalently real line bundles, over any space X are classified up to isomorphism by homotopy classes of maps [X, \mathbb{RP}^\infty].^[33] The tautological line bundle \gamma over \mathbb{RP}^\infty, whose fiber over each point \in \mathbb{RP}^\infty (representing a line l \subset \mathbb{R}^\infty) is the line l itself, is the universal real line bundle.^[33] For any real line bundle \xi over a space X, there exists a classifying map f: X \to \mathbb{RP}^\infty such that \xi \cong f^*\gamma, and the first Stiefel-Whitney class w_1(\xi) = f^*[\gamma] \in H^1(X; \mathbb{Z}_2) fully classifies \xi via this pullback. The Stiefel-Whitney classes of the universal line bundle \gamma satisfy w_0(\gamma) = 1, w_1(\gamma) = x (the generator of H^1(\mathbb{RP}^\infty; \mathbb{Z}_2)), and w_k(\gamma) = 0 for k > 1.^[33] In the broader context of orthogonal groups, \mathbb{RP}^\infty realizes the n=1 case of the infinite Grassmannian \mathrm{Gr}_n(\mathbb{R}^\infty), which serves as the classifying space BO(n) for rank-n real vector bundles.^[5] Thus, \mathbb{RP}^\infty = \mathrm{Gr}_1(\mathbb{R}^\infty) = BO(1), and the relation extends through the direct limit construction where finite approximations \mathbb{RP}^m \cong \mathrm{Gr}_1(\mathbb{R}^{m+1}) approximate the infinite case.^[5] The mod-2 cohomology ring H^*(\mathbb{RP}^\infty; \mathbb{Z}_2) \cong \mathbb{Z}_2 (with |x| = 1) generates the Stiefel-Whitney classes for line bundles as powers of the fundamental class x = w_1(\gamma), providing the polynomial algebra structure that underpins characteristic class computations for O(1)-bundles.^[5]

Applications in Topology

The infinite real projective space \mathbb{RP}^\infty serves as the classifying space for principal O(1)-bundles, equivalently real line bundles, over a paracompact base space B. Real line bundles over B are classified up to isomorphism by homotopy classes of maps [B, \mathbb{RP}^\infty], which correspond bijectively to elements of H^1(B; \mathbb{Z}_2).^[14] The first Stiefel-Whitney class w_1 \in H^1(B; \mathbb{Z}_2) of such a bundle is the primary characteristic class pulled back from the generator of H^1(\mathbb{RP}^\infty; \mathbb{Z}_2), and higher powers w_1^k generate the subring structure mirroring H^*(\mathbb{RP}^\infty; \mathbb{Z}_2) = \mathbb{Z}_2[w_1].^[14] For higher-rank principal O(n)-bundles, the classification extends via maps to the Grassmannian BO(n), with Stiefel-Whitney classes w_1, \dots, w_n \in H^*(B; \mathbb{Z}_2) satisfying relations that embed as a subring into the polynomial structure of H^*(BO; \mathbb{Z}_2), where the infinite-dimensional limit incorporates the line bundle case parameterized by \mathbb{RP}^\infty.^[14] In obstruction theory, \mathbb{RP}^\infty provides cohomological tools for analyzing sections and lifts due to its Eilenberg-MacLane space structure K(\mathbb{Z}_2, 1), with homotopy groups \pi_1(\mathbb{RP}^\infty) = \mathbb{Z}_2 and \pi_k(\mathbb{RP}^\infty) = 0 for k \geq 2. The primary obstruction to a nowhere-zero section of a real line bundle over a space X lies in H^1(X; \mathbb{Z}_2), identified with w_1, as this corresponds to the \pi_0-obstruction in the associated sphere bundle with fiber \mathbb{R} \setminus \{0\} \simeq S^0.^[14] Similarly, for lifting a map f: X \to \mathbb{RP}^\infty to its universal cover S^\infty, the primary obstruction resides in H^2(X; \pi_1(\mathbb{RP}^\infty)) = H^2(X; \mathbb{Z}_2), with higher obstructions vanishing due to the trivial higher homotopy groups.^[14] These obstructions detect non-triviality in bundle sections or double covers, linking directly to the mod-2 cohomology of the base. The relation between real K-theory and \mathbb{RP}^\infty arises through Adams operations on the groups KO^*(pt), which encode the stable homotopy of the orthogonal group spectrum. The connective real K-theory ring KO_0(pt) \cong \mathbb{Z} extends periodically with KO_n(pt) following Bott periodicity of period 8: KO_0(pt) = \mathbb{Z}, KO_1(pt) = \mathbb{Z}_2, KO_2(pt) = \mathbb{Z}_2, KO_3(pt) = 0, KO_4(pt) = \mathbb{Z}, and so on.^[34] Adams operations \psi^k: KO_0(X) \to KO_0(X) act as ring homomorphisms, satisfying \psi^k(L) = L^{\otimes k} on line bundles L and multiplicativity \psi^k \circ \psi^l = \psi^{kl}, allowing decomposition of virtual bundles into eigenspaces that relate KO^*(pt) to the completed cohomology \widehat{H}^*(\mathbb{RP}^\infty; \mathbb{Z}_2) = \mathbb{Z}_2[], where the completion captures infinite power series in the generator x \in H^1(\mathbb{RP}^\infty; \mathbb{Z}_2).^[34] This connection facilitates computations of KO^*(\mathbb{RP}^\infty) via the Atiyah-Hirzebruch spectral sequence, converging to the Adams-eigencomponent structure. Applications to embeddings and immersions of finite-dimensional real projective spaces \mathbb{RP}^n into Euclidean space \mathbb{R}^{n+k} leverage \mathbb{RP}^\infty as the parameter space for classifying the stable normal bundle \nu. For an immersion f: \mathbb{RP}^n \to \mathbb{R}^{n+k}, the normal bundle satisfies \tau_{\mathbb{RP}^n} \oplus \nu \cong \epsilon^{n+k}, the trivial bundle, so the classifying map \mathbb{RP}^n \to BO(k) for \nu is homotopic to the composite with the inclusion \mathbb{RP}^n \hookrightarrow \mathbb{RP}^\infty determining the stable tangent bundle's determinant line component.^[35] Obstructions to such immersions, such as the non-vanishing w_{2l-1}(\mathbb{RP}^{2l}) \neq 0 preventing immersion into \mathbb{R}^{4l-2}, arise from the Stiefel-Whitney classes of \nu pulled back from H^*(\mathbb{RP}^\infty; \mathbb{Z}_2), with \mathbb{RP}^\infty parameterizing the possible orientations and twisting in the stable range.^[35] This framework, via the Smale-Hirsch theorem, models the homotopy type of the immersion space as sections of a bundle over \mathbb{RP}^\infty in the limit.^[35]

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