On Maps with Unstable Singularities

If a continuous map f : X → Q is approximable arbitrary closely by embeddings X →֒ Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised by Ščepin and Akhmet’ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the answer is ‘generally no’ for (n,m) = (3, 6); (1, 3), and ‘yes’ when: 1) m > 2n, (n,m) 6= (1, 3); 2) m > 3(n+1) 2 and ∆(f) = {(x, y) | f(x) = f(y)} has an equivariant (with respect to the factor exchanging involution) mapping cylinder neighborhood in X ×X; 3) m > n+ 2 and f is the composition of a PL map and a TOP embedding. In doing this, we answer affirmatively (with a minor preservation) a question of Kirby: does small smooth isotopy imply small smooth ambient isotopy in the metastable range, verify a conjecture of Kearton–Lickorish: small PL concordance implies small PL ambient isotopy in codimension ≥ 3, and a conjecture set of Repovs–Skopenkov.