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

General

Proposition

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

(Hatcher 02, page 155 exercise 2)

(Hatcher 02, page 180)

Proposition

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

(Farber & Tabachnikov & Yuzvinsky 02, Theorem 12)

Proposition

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

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

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

(Farber & Tabachnikov & Yuzvinsky 02, Proposition 18)

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 $k \in \{0,\cdots, n\}$.

Homotopy groups

Proposition

(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 $S^0\rightarrow S^n\rightarrow\mathbb{R}P^n$ and are given by

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

Homology and cohomology

Proposition

(homology and cohomology of real projective space)

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

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

(Hatcher 02, Example 2.42)

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

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

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

(Hatcher 02, Corollary 3.28.)

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

The infinite real projective space $\mathbb{R}P^\infty \coloneqq \underset{\longrightarrow}{\lim}_n \mathbb{R}P^n$ is the classifying space $B O(1)$ 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

• quaternionic projective space

• octonionic projective space

• classifying space

References

See also:

• Topospaces, Real projective space
• Wikipedia, Real projective space

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

• Michael Farber, Serge Tabachnikov, Sergey Yuzvinsky, Topological robotics: motion planning in projective spaces (2002), arXiv:math/0210018

Homotopy groups, homology and cohomology of real projective space:

• Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

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)