The Isotopic Realization Problem
by
Peter M. Akhmet'ev
IZMIRAN
Coauthors: Sergey A. Melikhov (Moscow State University)

The Isotopic Realization Problem

Peter M. Akhmet'ev and Sergej A. Melikhov

Theorems of Haefliger, Weber and Harris reduce the problem of classification of embeddings M^m --> Nⁿ, n > [(3(m+1))/2] to a problem in homotopy theory. We extend this reduction to obtain solution of the Isotopic Realization Problem, which traces back to the works of Keldysh (1966) and Shchepin, Shtan'ko (1983).

Let f: M^m --> Nⁿ be a continuous map of a compact smooth manifold (compact space) into a smooth (topological) manifold. We call f discretely realizable if for arbitrary \epsilon > 0 there exists a smooth (topological) embedding g: M --> N such that ||g;f|| < \epsilon. We call f isotopically realizable if there exists a smooth (continuous) pseudo-isotopy H_t: N --> N, t in I, (i.e. a homotopy, which is a smooth (topological) isotopy for t in [0, 1), in particular H₀ is the identity) such that H₁ o g=f for some smooth (topological) embedding g: M --> N.

Isotopic Realization problem

Let f be a discretely realizable map. Is f isotopically realizable?

Remarks

1) For some wild knots f: S¹ --> R³ the answer is negative in DIFF category (Sikkema, 1972).

2) For maps of compact closed topological manifold onto itself the answer is positive in TOP category (Chernavskij, 1969 and Edwards, Kirby, 1971).

3) The answer is positive in case S^m --> S^m subset R^2m, m=4k+1 > 8 (Akhmet'ev, 1996).

4) For n > [(3(m+1))/2] the DIFF problem for a map f is equivalent to the TOP problem for f (Melikhov, unpublished).

5) For composition of a PL map and a TOP embedding the answer in TOP category is positive in codimension >= 3 and negative for (m, n)=(1, 3) (Melikhov, unpublished).

Theorem 1

If a map f: M^m --> Rⁿ is discretely realizable, where M is a compact polyhedron, n >= [(3(m+1))/2], then the map i o f: M^m --> Rⁿ subset Rⁿ⁺¹ is isotopically realizable.

Corollary

(by the first author) For each k there is such n that any map Sⁿ --> Sⁿ subset R^2n-k is isotopically realizable.

Theorem 2

For an arbitrary discretely realizable map f: M^m --> Nⁿ, where M is a compact polyhedron and N an orientable PL manifold, n > [(3(m+1))/2], there is a well-defined obstruction \beta such that f is isotopically realizable if and only if \beta = 0.

We present some examples of computation of the obstruction \beta.

Corollary

(by the second author) There is a map f: S¹×B²\sqcup B³ --> R⁶ (constructed explicitely) which is discretely realizable but not isotopically realizable.

Date received: May 27, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-15.