Editing Real projective space (section)

==Algebraic topology of real projective spaces==

===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>

===Homology===
The cellular chain complex associated to the above CW structure has 1 cell in each dimension 0, ..., ''n''. For each dimensional ''k'', the boundary maps ''d<sub>k</sub>'' : δ''D<sup>k</sup>'' → '''RP'''<sup>''k''−1</sup>/'''RP'''<sup>''k''−2</sup> is the map that collapses the equator on ''S''<sup>''k''−1</sup> and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2): 

<math display="block">\deg(d_k) = 1 + (-1)^k.</math>

Thus the integral [[cellular homology|homology]] is
<math display="block">H_i(\mathbf{RP}^n) = \begin{cases}
\mathbf{Z} & i = 0 \text{ or } i = n \text{ odd,}\\
\mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \text{odd,}\\
0 & \text{else.}
\end{cases}</math>

'''RP'''<sup>''n''</sup> is orientable if and only if ''n'' is odd, as the above homology calculation shows.