nLab real projective space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



The real projective space P n\mathbb{R}P^n is the projective space of the real vector space n+1\mathbb{R}^{n+1}.

Equivalently this is the Grassmannian Gr 1( n+1)Gr_1(\mathbb{R}^{n+1}).




Every continuous map P nP n\mathbb{R}P^n\rightarrow\mathbb{R}P^n for nn even has a fixed point. This does not hold for nn odd as in this case the continuous map P nP n,[x 0:x 1::x n1:x n][x 1:x 0::x n:x n1]\mathbb{R}P^n\rightarrow\mathbb{R}P^n, [x_0:x_1:\ldots:x_{n-1}:x_n]\mapsto[x_1:-x_0:\ldots:x_n:-x_{n-1}] does not have a fixed point.

(Hatcher 02, page 155 exercise 2)

(Hatcher 02, page 180)


For n1,3,7n\neq 1,3,7, the smallest number kk, so that there exists an immersion of real projective space P n\mathbb{R}P^n into cartesian space k1\mathbb{R}^{k-1}, is exactly the topological complexity TC(P n)\operatorname{TC}(\mathbb{R}P^n) (with convention TC(*)=1\operatorname{TC}(*)=1).

(Farber & Tabachnikov & Yuzvinsky 02, Theorem 12)


For n=1,3,7n=1,3,7 one has

TC(P n)=n+1 \operatorname{TC}(\mathbb{R}P^n) =n+1

for the topological complexity (with convention TC(*)=1\operatorname{TC}(*)=1).

(Farber & Tabachnikov & Yuzvinsky 02, Proposition 18)

Cell structure


(CW-complex structure)

For nn \in \mathbb{N}, the real projective space P n\mathbb{R}P^n admits the structure of a CW-complex.


Use that P nS n/(/2)\mathbb{R}P^n \simeq S^n/(\mathbb{Z}/2) is the quotient space of the Euclidean n-sphere by the /2\mathbb{Z}/2-action which identifies antipodal points.

The standard CW-complex structure of S nS^n realizes it via two kk-cells for all k{0,,n}k \in \{0, \cdots, n\}, such that this /2\mathbb{Z}/2-action restricts to a homeomorphism between the two kk-cells for each kk. Thus P n\mathbb{R}P^n has a CW-complex structure with a single kk-cell for all k{0,,n}k \in \{0,\cdots, n\}.

Homotopy groups


(homotopy groups of real projective space)

The homotopy groups of real projective plane can be calculated with the long exact sequence of homotopy groups (Hatcher 02, Theorem 4.41.) of the fiber bundle S 0S nP nS^0\rightarrow S^n\rightarrow\mathbb{R}P^n and are given by

(1)π k(P n)={* k=0 k=1,n=1 2 k=1,n>1 π k(S n) k1,n>0 \pi_k \big( \mathbb{R}P^n \big) \;=\; \left\{ \begin{array}{ll} \ast & k = 0 \\ \mathbb{Z} & k = 1, n = 1 \\ \mathbb{Z}_2 & k = 1, n > 1 \\ \pi_k \big( S^n \big) & k \geq 1, n > 0 \end{array} \right.

Homology and cohomology


(homology and cohomology of real projective space)

The ordinary homology groups of real projective space P n\mathbb{R}P^n can be calculated using its CW structure and are given by

(2)H k(P n)={ k=0ork=nifodd 2 koddand1kn 1 otherwise H_k \big( \mathbb{R}P^n \big) \;=\; \left\{ \begin{array}{ll} \mathbb{Z} & k = 0 \quad or \quad k = n \quad if \quad odd \\ \mathbb{Z}_2 & k \quad odd \quad and \quad 1 \leq k \leq n \\ 1 & otherwise \end{array} \right.

(Hatcher 02, Example 2.42)

Similarly the ordinary cohomology groups of P n\mathbb{R}P^n are

(3)H k(P n)={ k=0ork=nifodd 2 koddand1kn1 1 otherwise H^k \big( \mathbb{R}P^n \big) \;=\; \left\{ \begin{array}{ll} \mathbb{Z} & k = 0 \quad or \quad k = n \quad if \quad odd \\ \mathbb{Z}_2 & k \quad odd \quad and \quad 1 \leq k \leq n-1 \\ 1 & otherwise \end{array} \right.

One has H n1(P n) 2H_{n-1}(\mathbb{R}P^n)\cong\mathbb{Z}_2 für nn odd and H n1(P n)1H_{n-1}(\mathbb{R}P^n)\cong 1 for nn even, hence P n\mathbb{R}P^n is orientable iff nn is odd.

(Hatcher 02, Corollary 3.28.)

Relation to the /2\mathbb{Z}/2-classifying space

The infinite real projective space P lim nP n\mathbb{R}P^\infty \coloneqq \underset{\longrightarrow}{\lim}_n \mathbb{R}P^n is the classifying space B O ( 1 ) B O(1) for real line bundles. It has the homotopy type of the Eilenberg-MacLane space K(/2,1)=B/2K(\mathbb{Z}/2,1) = B \mathbb{Z}/2.

Kahn-Priddy theorem


See also:

On topological complexity of real projective space and connection with their immersion:

Homotopy groups, homology and cohomology of real projective space:

Computation of cohomotopy-sets of real projective space:

  • Robert West, Some Cohomotopy of Projective Space, Indiana University Mathematics Journal Indiana University Mathematics Journal Vol. 20, No. 9 (March, 1971), pp. 807-827 (jstor:24890146)

Last revised on March 12, 2024 at 02:50:52. See the history of this page for a list of all contributions to it.