Let $x_0 ∈ \Bbb C$ and $ S = \Bbb C \setminus \{x_0 \}$
$\forall x ∈ S$, prove existence and uniqueness of an isomorphism $ψ_{x_0, x} : π_1 (S, x) → \Bbb Z$ such that $ψ_{x_0 ,x} (α) = 1$, where $α$ is the homotopy class of the loop $κ : I → S$, based on $x$, defined by $t → x_0 + |x − x_0 | exp(i(2πt + arg(x − x_0 )))$
Some thoughts:
The problem is a mix between the index of a loop and the degree of an application that don't really understand.
$κ$ is a circle centered at $x_0$ with radius $|x-x_0|$ and $S$ has the same homotopy type as $κ$. so up to a translation from $0$ to $x_0$, there is a unique isomorphism $π_1 (S, x) \to \Bbb Z$ associating the homotopy class of $S$ to the degree of $α$.
Is the translation a valid argument to use the lifting of the exponential and reuse the degree application to define the index? If yes how to do it properly?
Thank you for any help and comments.