Contents

# Contents

## 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.

###### 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 $\operatorname{TC}(*)=1$).

###### Proposition

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

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

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

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

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

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

One has $H_{n-1}(\mathbb{R}P^n)\cong\mathbb{Z}_2$ für $n$ 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.

### 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 $K(\mathbb{Z}/2,1) = B \mathbb{Z}/2$.