# Real projective plane

## Definition

This is defined as the real projective space of dimension 2. Equivalently, it is the quotient of the 2-sphere by the equivalence relation that identifies antipodal (i.e., diametrically opposite) points.

It is denoted $\mathbb{R}\mathbb{P}^2$ or $\mathbb{P}^2\mathbb{R}$.

## Topological space properties

Property Satisfied? Is the property a homotopy-invariant property of topological spaces? Explanation Corollary properties satisfied/dissatisfied
manifold Yes No Based on the definition; in fact, any finite-dimensional real projective space is a manifold. We can also see this from the fact that its double cover, the 2-sphere, is a manifold satisfies: metrizable space, second-countable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc.
path-connected space Yes Yes Can be seen directly, or from the fact that its double cover, the 2-sphere, is path-connected. satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous)
simply connected space No Yes It has a double cover, namely the 2-sphere, which is path-connected. In fact, the double cover is simply connected, so the fundamental group of the space is a cyclic group of order two. dissatisfies: weakly contractible space, contractible space
acyclic space No Yes $H_1$ of the space is cyclic of order two (this can be seen from the Hurewicz theorem and the fact that $\pi_1$ is cyclic of order two, or directly using the homology of real projective space). dissatisfies: weakly contractible space, contractible space
rationally acyclic space Yes Yes All the homology groups (other than the zeroth homology group) are torsion, so the rational homology groups are all zero. space with Euler characteristic one
compact space Yes No Its double cover, the 2-sphere, is compact, and compactness is continuous image-closed dissatisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness

## Algebraic topology

### Homology groups

Further information: homology of real projective space

The homology groups with coefficients in $\mathbb{Z}$ are as follows: $H_0(\mathbb{R}\mathbb{P}^2) \cong \mathbb{Z}$, $H_1(\mathbb{R}\mathbb{P}^2) \cong \mathbb{Z}/2\mathbb{Z}$, and all higher homology groups are zero. In particular, the second homology group is zero, which can be explained by the non-orientability of the real projective plane. For more information, see homology of real projective space.

### Cohomology groups

Further information: cohomology of real projective space

The cohomology groups with coefficients in $\mathbb{Z}$ are as follows: $H^0(\mathbb{R}\mathbb{P}^2) \cong \mathbb{Z}$, $H^1(\mathbb{R}\mathbb{P}^2) = 0$, $H^2(\mathbb{R}\mathbb{P}^2) \cong \mathbb{Z}/2\mathbb{Z}$, and all higher cohomology groups are zero. The cohomology ring is $\mathbb{Z}[x]/(x^2,2x)$, where $x$ is the non-identity element of $H^2$.

### Invariants based on homology

These are all invariants that can be computed in terms of the homology groups.

Invariant General description Description of value for real projective space $\R\mathbb{P}^n$ Description of value for $\R\mathbb{P}^2$
Betti numbers The $k^{th}$ Betti number $b_k$ is the rank of the torsion-free part of the $k^{th}$ homology group. $b_0 = 1$. $b_n = 1$ if $n$ is odd and $b_n = 0$ if $n$ is even. $b_0 = 1$, all other $b_k = 0$.
Poincare polynomial Generating polynomial for Betti numbers $1 + x^n$ if $n$ is odd. $1$ if $n$ is even. 1
Euler characteristic $\sum_{k=0}^\infty (-1)^k b_k$ $0$ if $n$ is odd. $1$ if $n$ is even. Note that the Euler characteristic is half the Euler characteristic of the sphere $S^n$, which is its double cover. 1 (in this case, half the Euler characteristic of the 2-sphere $S^2$)

### Homotopy groups

Further information: covering map induces isomorphisms on higher homotopy groups, homotopy of spheres

The quotient map $S^2 \to \mathbb{R}\mathbb{P}^2$ is a universal covering map. In particular, this map induces isomorphisms on all $\pi_k, k \ge 2$. Further, since the map is a double cover, $\pi_1(\mathbb{R}\mathbb{P}^2) \cong \mathbb{Z}/2\mathbb{Z}$. We thus conclude that:

$n$ Common name for $\pi_n$ Value of $\pi_n(\mathbb{R}\mathbb{P}^2)$ Explanation
0 set of path components one-point space it is a path-connected space
1 fundamental group $\mathbb{Z}/2\mathbb{Z}$ its universal cover $S^2$ is a double cover
2 second homotopy group $\mathbb{Z}$ same as $\pi_2(S^2)$, since the covering map $S^2 \to \R\mathbb{P}^2$ induces an isomorphism on all higher homotopy groups
3 third homotopy group $\mathbb{Z}$ same as $\pi_3(S^2)$, which is generated by the Hopf fibration
4 fourth homotopy group $\mathbb{Z}/2\mathbb{Z}$ same as $\pi_4(S^2)$ (?)