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 $$\mathbb{P}^2\mathbb{R}$$.

Homology groups
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
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.

Homotopy groups
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: