Fundamental group of $\mathbb{P}^n(\mathbb{C})$

Question

We know that $$\Pi_1(\mathbb{P}^n(\mathbb{R})) \cong \mathbb{Z}_2 $$ for $n \geq 2 $.

Is there a similar statement for $\Pi_1(\mathbb{P}^n(\mathbb{C}))$ ?

Answer 1

The space $X=\mathbb P^n(\mathbb C)$ has a CW-structure $\mathbb P^n(\mathbb C)=e^0\cup e^2\cup\cdots \cup e^{2n}$ with just one open cell in each even dimension.
The skeletons are $X_{2k}=X_{2k+1}=\mathbb P^{k}(\mathbb C)\quad (0\leq k\leq n)$.
But in all generality we have for a connected CW-complex $\pi_1(X)=\pi_1(X_2)$ so that in our case $$\pi_1(\mathbb P^n(\mathbb C))=\pi_1(X)=\pi_1(X_2)=\pi_1(\mathbb P^1(\mathbb C))=\pi_1 (S^2)=0$$ Thus all spaces $\mathbb P^n(\mathbb C)$ are simply connected.