Intereting Posts

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?
Upper bounds on the size of $\operatorname{Aut}(G)$
Natural isomorphism between $\mathbb{C}\otimes_{\mathbb{R}}V$ and $V\oplus V.$
Does Fermat's Last Theorem hold for cyclotomic integers in $\mathbb{Q(\zeta_{37})}$?
What is the status of the purported proof of the ABC conjecture?
Finding the null space of a matrix by least squares optimization?
Visualizing Lie groups.
General McNugget problem
Obstructions to lifting a map for the Hopf fibration
Prove that the equation $x^{10000} + x^{100} – 1 = 0$ has a solution with $0 < x < 1$
Is the union of finitely many open sets in an omega-cover contained within some member of the cover?
symmetric positive definite matrix question
Counting subsets containing three consecutive elements (previously Summation over large values of nCr)
Growth-rate vs totality
Do the digits of $\pi$ contain every possible finite-length digit sequence?

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?

- The First Homology Group is the Abelianization of the Fundamental Group.
- Equivalence of knots: ambient isotopy vs. homeomorphism
- What does 2v mean in the context of Simplicial Homology
- Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.
- The complement of Jordan arc
- Are there any non-trivial group extensions of $SU(N)$?
- is the group of rational numbers the fundamental group of some space?
- Isomorphism of varieties via coordinate rings
- Relation between Stiefel-Whitney class and Chern class
- On the definition of projective vector bundle.

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.

- When do the Freshman's dream product and quotient rules for differentiation hold?
- The Isomorphism of a Linear Space with Its Dual and Double Dual
- (Computationally) Simple sigmoid
- Where is the flaw in my Continuum Hypothesis Proof?
- $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed
- How to represent each natural number?
- Proof about dihedral groups
- Construction of a function which is not the pointwise limit of a sequence of continuous functions
- For which $\mathcal{F} \subset C$ does there exist a sequence converging pointwise to the supremum?
- Finite dimensional algebra with a nil basis is nilpotent
- example of discontinuous function having direction derivative
- Problems and Resources to self-study medium level math
- Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$
- Prove that the countable complement topology is not meta compact?
- Proof that Right hand and Left hand derivatives always exist for convex functions.