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}).


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\}.

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 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:

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 November 7, 2023 at 13:15:38. See the history of this page for a list of all contributions to it.