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 Akhmetʹev. 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: gt;2n, (n,m)≠(1,3); gt;3(n+1)/2 and Δ(f)={(x,y)∣f(x)=f(y)} has an equivariant (with respect to the factor exchanging involution) mapping cylinder neighborhood in X×X; gt;n+2 and f is the composition of a PL map and a TOP embedding. ng 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.