Fundamental group of multiplicative group in Zariski topology

Question

What is the fundamental group of the multiplicative group of the complex numbers $\mathbb{G}_m(\mathbb{C})$ with respect to the Zariski topology. More precisely, what are the homotopy classes of continuous loops $f:[0,1]\rightarrow \mathbb{G}_m(\mathbb{C})$ with a fixed base point?

Answer 1

The Zariski topology on $\mathbb{C}^{\ast}$ is the cofinite topology, so a map into $\mathbb{C}^{\ast}$ is continuous if and only if the preimage of every point is closed. (This is an extremely general class of maps.) If $b \in \mathbb{C}^{\ast}$ is a basepoint and $f, g : [0, 1] \to \mathbb{C}^{\ast}$ are two maps based at $b$, consider the homotopy

$$H(x, t) = \begin{cases} f(x) & \text{ if } t = 0 \\ g(x) & \text{ if } t = 1 \\ b & \text{ if } x = 0, 1 \\ \text{arb}(x, t) & \text{ otherwise} \end{cases}$$

where $\text{arb}(x, t)$ is an arbitrary bijection from whatever part of the domain hasn't already been covered to $\mathbb{C}^{\ast}$. Then the preimage $H^{-1}(y)$ of any point which is not $b$ is the union of a point and two closed sets, which is closed, and the preimage of $b$ is the union of four closed sets, so $H$ is continuous. ($H$ is not much of a homotopy, but then, the cofinite topology is not much of a topology.)

In other words, $\pi_1(\mathbb{C}^{\ast})$ is trivial.

But this is almost certainly not what you want. Taking the topological fundamental group with respect to the Zariski topology is not a good notion of fundamental group for varieties. You should be either taking the topological fundamental group with respect to the analytic topology or taking the étale fundamental group.