All Questions

Suppose that $X, B$ are smooth irreducible varieties over $\mathbb{C}$ and $f : X \to B$ is a smooth proper morphism. Then we can consider the homotopy exact sequence: $$ \pi_2(B) \to \pi_1(F) \to \...

Does there exist a complex smooth proper variety whose fundamental group is finite non-abelian simple?

I would like to know: What does the fundamental group of a quasi-projective algebraic variety look like? I remember that I have seen somewhere that for a connected, finite-type CW-complex $X$, ...

Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois ...

This is another vague question. Hope you guys don't mind. Let $T$ be a Tannakian category. For any fibre functor $F$ on $T$ we define the fundamental group of $T$ at $F$, denoted by $\pi_1(T,F)$, to ...