On isotopic realizability of continuous maps
by
Sergey A. Melikhov
Moscow State University

Let f: X --> Q be a continuous map of a compact space into a manifold. We call f discretely realizable if for any \epsilon > 0 there exists an embedding g: X --> Q, \epsilon-close to f. We call f isotopically realizable if there exists a pseudo-isotopy H_t:Q --> Q, t in [0, 1], (i.e. a homotopy which is the identity for t=0 and a homeomorphism for t < 1) such that H₁ o g=f for some embedding g: X --> Q.

Of course, isotopic realizability implies discrete one; when the converse holds? This question, known as Isotopic Realization Problem, traces back to the papers of Keldysh (1966) and Shchepin, Shtan'ko (1983), and was explicitely stated by Shchepin and Akhmet'ev in 1996. It is known that discrete realizability implies isotopic for any map

1) of a compact closed TOP manifold onto itself (Chernavskij, 1969; Edwards, Kirby, 1971);

2) S^m --> S^m subset R^2m, m=4k+1 >= 9 (Akhmet'ev, 1996);

3) S^m --> S^m subset R^2m-k, k arbitrary, m=m(k) (Akhmet'ev, 1998);

4) of a compact n-polyhedron into an orientable PL m-manifold, m > 3(n+1)/2, iff a cohomological obstruction vanishes (Akhmet'ev, Melikhov).

Theorem There exist discretely realizable but not isotopically realizable maps

a) I¹\sqcup I¹ --> R³ (with one double point) and

b) I³\sqcup S¹×I² --> R⁶ (with singular set a point plus the 3-adic solenoid).

Theorem Let Xⁿ be a compact polyhedron, Q^m a PL manifold without boundary (e.g. Euclidean space) and f: X --> Q a discretely realizable map. Then f is isotopically realizable, provided either

a) m-n >= 3 and f is composition of a PL map and a TOP embedding, or

b) m >= 2n+1, (m, n) =/= (3, 1).

Date received: June 9, 1999