Is complex projective space simply connected?

Question

I know real projective space isn't simply connected, what about complex projective spaces?

Answer 1

Yes. One of many ways to see this is to fit $\mathbb{CP}^n$ into a fiber sequence

$$S^1 \to S^{2n+1} \to \mathbb{CP}^n$$

(since $S^1 \cong \text{U}(1)$ acts by scalars on the unit sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$ with quotient $\mathbb{CP}^n$) and apply the long exact sequence in homotopy. The long exact sequence also shows that $\pi_2(\mathbb{CP}^n) \cong \mathbb{Z}$, exactly as one would expect from the Hurewicz theorem due to the fact that $H_2(\mathbb{CP}^n) \cong \mathbb{Z}$, but after that the homotopy groups are boring for awhile until they become the homotopy groups of $S^{2n+1}$. For $n = 1$ the fiber sequence above is the Hopf fibration and this recovers the fact that the higher homotopy of $S^3$ and $S^2$ agree.