Constructing near-embeddings of codimension one manifolds with countable dense singular sets

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\sl near-embedding}.

💡 Research Summary

The paper addresses a long‑standing question in geometric topology concerning the relationship between the size of the singular set of a continuous map and the possibility of approximating that map by an embedding (a “near‑embedding”). The authors focus on maps of codimension one, i.e., continuous maps (f\colon M^{n-1}\to M^{n}) where both source and target are closed manifolds and (n\ge 3).

A conjecture (Conjecture 1.1) proposed in the mid‑1980s asserts that if the singular set (S(f)={x\in M^{n-1}\mid f^{-1}(f(x))\neq{x}}) has topological dimension zero, then for every (\varepsilon>0) there exists an embedding (g) such that (\sup_{x} d(f(x),g(x))<\varepsilon). In dimension three this conjecture is equivalent to the Bing conjecture and is closely tied to the 3‑dimensional recognition problem. Earlier work proved the conjecture under the additional hypothesis that the closure of the singular set is also zero‑dimensional; however, examples where the closure has larger dimension required highly technical constructions.

The contribution of this paper is a remarkably simple construction that works for every (n\ge 3) and produces a map whose singular set is countable, dense (hence non‑closed) in the source, yet the map is still a near‑embedding. The construction proceeds as follows.

1. Choice of a countable basis.
   The authors fix a countable basis ({U_i}_{i\in\mathbb N}) of open subsets of the standardly embedded sphere (S^{n-1}\subset S^{n}).

2. Inductive insertion of tame arcs.
   They inductively build a sequence of pairwise disjoint tame PL arcs (\alpha_i\subset S^{n}) with two key properties:

   • The endpoints of (\alpha_i) lie in (U_i) and are exactly (\partial\alpha_i\subset S^{n-1}).
   • The diameter of (\alpha_i) is less than (2^{-i}).
     Because the diameters tend to zero, the family ({\alpha_i}) forms a null‑sequence.

3. Cellular decomposition.
   The collection (\mathcal G={\alpha_i\mid i\in\mathbb N}) together with the remaining points of (S^{n}) defines a cellular decomposition of (S^{n}). Each element of (\mathcal G) is the intersection of a nested sequence of closed (n)‑cells, and the decomposition is upper semicontinuous. By a theorem of Daverman (Theorem 7 in his book on decompositions), such a decomposition is shrinkable: the quotient map (\pi\colon S^{n}\to S^{n}/\mathcal G) can be approximated arbitrarily closely by homeomorphisms. Moreover, the quotient space (S^{n}/\mathcal G) is homeomorphic to the original sphere (S^{n}).

4. Definition of the map.
   Let (i\colon S^{n-1}\hookrightarrow S^{n}) be the inclusion. The desired map is defined as the composition
   \