Editing Real projective space (section)

===Homotopy groups===
The higher homotopy groups of '''RP'''<sup>''n''</sup> are exactly the higher homotopy groups of ''S<sup>n</sup>'', via the long exact sequence on homotopy associated to a [[fibration]].

Explicitly, the fiber bundle is: <math display="block">\mathbf{Z}_2 \to S^n \to \mathbf{RP}^n.</math>
You might also write this as <math display="block">S^0 \to S^n \to \mathbf{RP}^n</math>
or <math display="block">O(1) \to S^n \to \mathbf{RP}^n</math> by analogy with [[complex projective space]].

The homotopy groups are:
<math display="block">\pi_i (\mathbf{RP}^n) = \begin{cases}
0 & i = 0\\
\mathbf{Z}   & i = 1, n = 1\\
\mathbf{Z}/2\mathbf{Z} & i = 1, n > 1\\
\pi_i (S^n) & i > 1, n > 0.
\end{cases}</math>