Isotopic Realization of Continuous Mappings

Abstract

In the metastable range, a class of mappings yielding a negative solution of the isotopic realization problem (posed by E. V. Shchepin in 1993) and satisfying an additional technical condition is described in algebraic terms. Namely, one constructs an obstruction to isotopic realization of a discretely realizable continuous mapping f of an n-polyhedron to an orientable PL m-manifold; the completeness of this obstruction is established for $m >\frac{{3(n + 1)}}{2}$ in the case where f is discretely realizable by skeleta. Also, for $n \geqslant 3$, a series of mappings $S^{ n} \to \mathbb{R}^{2n}$ (with singular set consisting of a p-adic solenoid, $p \geqslant 3$, and a point) is presented for which the problem is solved in the negative. Furthermore, it is shown that the problem is solved in the affirmative in the metastable range if stabilization with codimension one is allowed, as well as in the case of a mapping $f:S^n \to \mathbb{R}^m$, under the condition that f is discretely realizable by skeleta and the configuration singular set $\sum (f) = \{ (x,y) \in S^n \times S^n |f(x) = f(y)\}$ is acyclic in dimension $2n - m$ (in the sense of the Steenrod―Sitnikov homology). Bibliography: 31 titles.