Links modulo knots and the isotopic realization problem (Q2784563)

Let \(f\) be a continuous map from a compact polyhedron \(X\) to a manifold \(Q\). Does discrete realizability of \(f\), that is, \(\varepsilon\)-approximability by embeddings for all \(\varepsilon>0\), imply isotopic realizability, that is, the existence of a pseudo-isotopy \(H_t:Q\to Q\), \(t\in I=[0,1]\) (a homotopy that realizes a homeomorphism for \(t<1)\) transferring to \(f\) some embedding \(g:X \hookrightarrow Q\) (which means that \(H_0=\text{id}\) and \(H_1g=f)\)? This question, which dates back to Keldysh's problem (1966) on the realization of wildly embedded polyhedra by pseudo-isotopic subpolyhedra, was posed by E. V. Shchepin in 1993 and is known as the isotopic realization problem.NEWLINENEWLINENEWLINEFollowing the first author's paper [On maps with unstable singularities, Topology Appl. 120, No. 1-2, 105-156 (2002; Zbl 1013.57016)], the authors discuss some approaches to the question whether there exist locally flat topological immersions \(S^1\sqcup S^1\to \mathbb{R}^3\) that are discretely realizable but not isotopically realizable.