Let the set $S$ be defined as: $$S = \{(a_{1},a_{2} \dots, a_{n}) \mid \Sigma^{n}_{1}a^{2}_{i} = 1, a_{n} = 0\}$$ Then, let $A = \mathbb{R}^{n} - S$. Intuitively, it seems clear to me that $\pi_{1}(A,c) \cong \mathbb{Z}$, where $c$ is an arbitrary element of $A$. If this is true, how can I distinguish between curves in $A$ that are not homotopic to each other? Is there something like the crossing number that can generalize to higher dimensions that I can use for this?
For the sake of reference, my intuitions are based primarily on the arguments offered page 43 of hatcher, under the section on linking circles.
$X$ is the union of two simply connected open subspaces $X_1,X_2$. WLOG we can suppose $x \in X_1$. A loop with $x$ for base point is included in $X_1$ and homotopic to the constant loop equal to $x$. Therefore $\pi_1(X,x)$ is the trivial group.
Am I missing something?