Real projective three-dimensional space

Definition
As a topological space, this can be defined in the following equivalent ways:


 * The defining ingredient::real projective space of dimension 3. It is denoted $$\R\mathbb{P}^3$$ or $$\mathbb{P}^3\R$$. In particular, it is the quotient of the defining ingredient::3-sphere under the antipodal map.
 * The underlying topological space of the defining ingredient::special orthogonal group $$SO(3,\R)$$.

Note that real projective spaces are not in general homeomorphic to the underlying spaces of special orthogonal groups.

Homology groups
The homology groups with coefficients in integers are as follows:

$$H_p(\R\mathbb{P}^3; \mathbb{Z}) = \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0,3 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p = 1 \\ 0 & \qquad p = 2 \ \operatorname{or} \ p > 3 \\\end{array}$$

The top homology group is $$\mathbb{Z}$$, indicating that the manifold is orientable, which it is. This is common to all odd-dimensional real projective spaces.

The homology groups with coefficients in a module $$M$$ over a ring are as follows:

$$H_p(\R\mathbb{P}^3; M) = \lbrace \begin{array}{rl} M, & \qquad p = 0,3 \\ M/2M, & \qquad p = 1 \\ T & \qquad p = 2 \\ 0 & \qquad p > 3 \\\end{array}$$

Here, $$T$$ denotes the 2-torsion submodule of $$M$$. In particular, if $$M$$ is uniquely 2-divisible (the case for any field of characteristic not equal to 2), we get:

$$H_p(\R\mathbb{P}^3; M) = \lbrace \begin{array}{rl} M, & \qquad p = 0,3 \\ 0, & \qquad \operatorname{otherwise} \\\end{array}$$

Cohomology groups
The cohomology groups with coefficients in integers are as follows:

$$H^p(\R\mathbb{P}^3; \mathbb{Z}) = \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0,3 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p = 2 \\ 0 & \qquad p = 1 \ \operatorname{or} p > 3 \\\end{array}$$

These can be obtained from the homology groups using the Poincare duality theorem, among other methods.

The cohomology groups with coefficients in a module $$M$$ over a ring are as follows:

$$H^p(\R\mathbb{P}^3; M) = \lbrace \begin{array}{rl} M, & \qquad p = 0,3 \\ M/2M, & \qquad p = 2 \\ T & \qquad p = 1 \\ 0 & \qquad p p > 3 \\\end{array}$$

Here, $$T$$ denotes the 2-torsion submodule of $$M$$. In particular, if $$M$$ is uniquely 2-divisible (the case for any field of characteristic not equal to 2), we get:

$$H_p(\R\mathbb{P}^3; M) = \lbrace \begin{array}{rl} M, & \qquad p = 0,3 \\ 0, & \qquad \operatorname{otherwise} \\\end{array}$$