nLab
real projective space

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

Revised on June 11, 2017 10:44:58 by Urs Schreiber (88.77.226.246)