nLab real projective space

Contents

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

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}).

Properties

Cell structure

Proposition

(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.

Proof

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 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

References

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 October 27, 2021 at 06:25:53. See the history of this page for a list of all contributions to it.