everyone! I'm a physics PhD student who knows little about algebraic topology. Recently I encountered a problem about the topology of the manifold $$M:=\left\{ (z_1,z_2,z_3)\in\mathbb{C}^3 \mid z_1^2+z_2^2+z_3^2\neq 0 \right\}. $$ where I need to calculate the fundamental group of $M$.
My first step is to consider the compliment set $$\overline{M}=\left\{ (z_1,z_2,z_3)\in\mathbb{C}^3\mid z_1^2+z_2^2+z_3^2= 0 \right\}. $$ Because the equation $z_1^2+z_2^2+z_3^2= 0$ is a homogeneous equation, the set $\overline{M}$ is a star-shaped set.
Next, I change the complex variables to the real ones, i.e. \begin{align*} z_1 &= x_1 + iy_1 \\ z_2 &= x_2 + iy_2 \\ z_3 &= x_3 + iy_3 \\ \end{align*} and the complex equation $z_1^2+z_2^2+z_3^2= 0$ is equivalent to the real system of equations \begin{align} \left(x_{1}\right)^{2}+\left(x_{2}\right)^{2}+\left(x_{3}\right)^{2} & =\left(y_{1}\right)^{2}+\left(y_{2}\right)^{2}+\left(y_{3}\right)^{2} \label{eq1}\\ x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3} & =0 \label{eq2} \end{align}
Because the set $\overline{M}$ is a star-shaped set, I can move the point in $M$ along the rays started from the origin to the unit sphere $S^5$, so that $M$ is homotopic equivalent to $S^5 - \overline{M}\cap S^5$. Next, I consider the two real vectors $\mathbf{x}=(x_1,x_2,x_3)$ and $\mathbf{y}=(y_1,y_2,y_3)$. When the 6-vector $(\mathbf{x},\mathbf{y})$ is restricted on $S^5$, the 3-vectors $\mathbf{x}$ and $\mathbf{y}$ are restricted on $S^2$ respectively, and an additional restriction $\mathbf{x}\cdot \mathbf{y}=0$ is required, which restricts $\mathbf{y}$ on the equator perpendicular to $\mathbb{x}$. Therefore, the set $\overline{M}\cap S^5$ is locally homeomorphic to $S^2\times U(1)$, which means that it is an $U(1)$-principal bundle on $S^2$, or the bundle of all the unit tangent vectors of a 2D sphere. However, I don't know whether it is globally homeomorphic to $S^2\times U(1) $. Then, I'm stuck because I don't know how to calculate the fundamental group of this guy.
My questions are the followings:
- Are there any general methods or theorems that deal with the fundamental group of $\mathbb{R}^n$ subtracted by a star-shaped domain?
- If not, are there any methods to solve my problem, that is, the fundamental group of $S^5$ substracted by the bundle of the unit tangent vectors on $S^2$?
Maybe the problem is similar to the fundamental group of $S^3-S^1$ ?
