Suppose $\{U_s \mid s \in (0,1)\}$ is a continuous nested collection of bounded Euclidean domains, i.e. each $U_t$, $t \in (0,1)$ is a bounded domain in $\mathbb{R}^n$ such that $U_s \subset U_t$ whenever $s \leq t$, and the mapping $t \mapsto U_t$ is continuous with respect to the Hausdorff metric (recall that the domains are bounded and thus their closures are compact). Suppose further that there exists a constant $k \in \mathbb{N}$ such that $\# \pi_1(U_s) = k$ for all $s \in (0,1)$.
Does there exist a parameter $t_0 \in (0,1)$ such that $\pi_1(U_s) \simeq \pi_1(U_t)$ for all $t,s \in (t_0,1)$?
Further notes:
- We do not assume that the diameter of the nested domains is tending to zero.
- If needed we may further assume that each of the domains $U_t$ has exactly two boundary components, and in fact all of the domains share one of the boundary components. (So these are some sort of generalized annuli converging towards the external boundary sphere.)
- In dimension three the claim is trivial as spatial Euclidean domains have torsion free fundamental groups.
EDIT: I added a crucial missing assumption on the continuity of the nested collection.
Still I think that a ball can become a torus and viceversa; intuitively north and south pole approaches until they meet (and we still have trivial fundamental group); then the meeting point becomes a hole and we get a full torus. This is continuous wrt the Hausdorff metric and creates / erases holes; the problem is the same, of finding two things with different finite group structure with the same cardinality. Do you have eg some examples of Z/2Z x Z/2Z and Z/4Z?