nLab real projective space

Idea

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

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

Properties

Cell structure

Proposition

(CW-complex structure)

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

Proof

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

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

Relation to classifying space

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

Kahn-Priddy theorem

• Kahn-Priddy theorem

Related concepts

• complex projective space

• classifying space

References

See also:

• Wikipedia, 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)