Calculate the homotopy groups of $\mathbb{C}P^{\infty}$.
And I have a hint there is a fibre sequence $S^1 \rightarrow S^{\infty} \rightarrow \mathbb{C}P^{\infty}.$
But still I do not know how to calculate homotopy groups, Could anyone help me in this please?
$S^{\infty}$ is contractible, the Serre exact sequence gives $\pi_{n+1}(\mathbb{C}^{\infty})=\pi_n(S^1)$.
https://mathoverflow.net/questions/198/how-do-you-show-that-s-infty-is-contractible
https://en.wikipedia.org/wiki/Homotopy_group#Long_exact_sequence_of_a_fibration