On isotopic realizability of continuous mappings

Under the metastable dimension restriction, we present an algebraic description of the class of discretely realizable maps (i.e., maps that are arbitrarily closely approximable by embeddings) which fail to be isotopically realizable (i.e., to be obtained from an embedding via a pseudo-isotopy). These maps are precisely the ones which yield a negative solution to the Isotopic Realization Problem of E. V. Shchepin (1993), see [1, 27].
A cohomological obstruction for isotopic realizability of a discretely realizable map of an $n$-polyhedron into an orientable PL $m$-manifold is constructed. We also present an obstruction for discrete realizability of a map $S^n\to\mathbb R^m$. If $m>\frac{3(n+1)}2$, these obstructions are shown to be complete. In fact, the latter obstruction can be regarded as an element of the limit of certain inverse spectrum of finitely generated Abelian groups (which are cohomology groups of compact polyhedra with coefficients in a locally constant sheaf), while the first obstruction can be identified with an element of the derived limit of this spectrum. On the other hand, the obstructions generalize the classical van Kampen obstruction for embeddability of an $n$-polyhedron into $\mathbb R^{2n}$.
An explicit construction of a series of discretely but not isotopically realizable maps $S^n\to\mathbb R^{2n}$ is given for $n\geqslant3$. The singular sets of these maps are homeomorphic to the disjoint union of the $p$-adic solenoid, $p\geqslant3$, and a point. Furthermore, it is shown that the Isotopic Realization Problem has positive solution in the metastable range under the assumption of stabilization with codimension 1, or if the configuration singular set $\Sigma(f)=\{(x,y)\in S^n\times S^n\mid f(x)=f(y)\}$ of a map $f\colon S^n\to\mathbb R^m$ is acyclic in dimension $2n-m$ with respect to the Steenrod–Sitnikov homology.