I need to compute the fundamental group of $X = \Bbb R^n\setminus\Bbb S^1$, for $n\ge 3$. My teacher has provided me with a previous step, which is by using S-VK theorem on these two open sets: \begin{equation} A = X\setminus\{(x_1,...,x_n) \in R^n : x_1=...=x_{n-1}=0\}, \end{equation} which is just $\Bbb R^n\setminus(\Bbb S^1\cup\{\text{axis}\})$, and \begin{equation} B = \{(x_1,...,x_n) \in X : x_1^2+...+x_{n-1}^2 < 1\}, \end{equation} which is contractible, since it is the interior of a "(n-1)-dimensional cylinder".
If I rewrite the first subspace like $A = T\times \Bbb S^1$, for \begin{equation} T = \{(x_1,...,x_{n}) \in X : x_1>0, x_2=0\}, \end{equation} I get that the fundamental group of $A$ is \begin{equation} \pi_1(A) = \begin{cases} \Bbb Z^2 & \quad\text{for $n\in\{3,4\}$} \\ \Bbb Z & \quad \text{for $n>4$} \end{cases} \end{equation} since for $n\ge 4$, $T$ is homeomorphic to $\Bbb R^{n-1}\setminus\text{\{straight line\}}$, and for $n=3$, $T\simeq \Bbb S^1$, so \begin{equation} \pi_1(T) = \begin{cases} \Bbb Z & \quad\text{for $n\in\{3,4\}$} \\ \Bbb \{0\} & \quad \text{for $n>4$} \end{cases} \end{equation}
It is also clear that $A\cap B\simeq \Bbb S^1$, then $\pi_1(A\cap B)=\Bbb Z$.
I want to use a result (taken from the S-VK theorem for 2 open sets) which states that if $B$ is 1-connected, then $\pi_1(X) = \pi_1(A)/N$, where $N$ is the least normal subgroup that contains $Im(i_1)_*$, and $(i_1)_*:\pi_1(A\cap B)\to\pi_1(A)$.
So, I have two questions:
Did I do the fundamental group of $T$ correctly? It seems odd, but from this post I know that the case for $n=3$ is at least correct. Is it, though, for $n\ge 4$?
How do I compute $N$? I'm a bit green in algebraic topology, and I cannot find many examples on how to compute this subgroup $N$ out of data from the subspaces.
I already know the answer of $\pi_1(\Bbb R^n\setminus\Bbb S^1)$, which is answered in this other post, but I need to reach the answer through this previous step. Could anyone please help me? Thanks in advance!