Homotopy groups of $\mathbb{RP}^\infty$, $\mathbb{CP}^\infty$.

Question

Could someone supply me a precise reference to the computation of all homotopy groups of infinite real projective space and infinite complex projective space?

Answer 1

Because $S^\infty$ is contractible, and we have a fiber bundles $S^1 \to S^\infty \to \Bbb{CP}^\infty$, the long exact sequence of homotopy groups of a pair shows that its only nonvanishing homotopy group is $\pi_2(\Bbb{CP}^\infty) = \Bbb Z$. See Hatcher, 4.50. Example 4.44 is a construction of this bundle.

$\Bbb{RP}^\infty$ is more elementary: if $\tilde M$ is a cover of $M$, then $\pi_i(\tilde M) = \pi_i(M)$ for all $i>1$. So because the contractible $S^\infty$ double covers $\Bbb{RP}^\infty$, it has fundamental group $\Bbb Z/2$ and no other nonzero homotopy groups. See Hatcher, 1B.3.