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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.

Rami Luisto
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    I think that there could be a sort of cyclic situation, where two different objects alternate (change given at some critical point $t_n$). I don't see any obstruction a priori to some holes appearing and disappearing in a "orchestrated" way so that conditions are satisfied. Problem with finding a counterexample is to find X,Y with different fundamental groups of the same size, that embeds in the same R^n, in two different ways. In the first way, X is in Y; in the second, Y is in X. For example, think about a ball and a full torus: one can emb d one into another. – Andrea Marino Sep 12 '19 at 12:59
  • Oh that's a good point, I forgot to add one critical assumption on the continuity of the collection. Thanks! – Rami Luisto Sep 12 '19 at 13:23

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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?

Andrea Marino
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