On isotopic realizability of maps factored through a hyperplane

In this paper we study the isotopic realization problem, which is the question of isotopic realizability of a given (continuous) map $f$, that is, the possibility of a uniform approximation of $f$ by a continuous family of embeddings $g_t$, $t\in[0,\infty)$, under the condition that $f$ is discretely realizable, that is, that there exists a uniform approximation of $f$ by a sequence of embeddings $h_n$, $n\in\mathbb N$.
For each $n\geqslant3$ a map $f\colon S^n\to\mathbb R^{2n}$ is constructed that is discretely but not isotopically realizable and which, unlike all such previously known examples, is a locally flat topological immersion. For each $n\geqslant4$ a map $f\colon S^n\to\mathbb R^{2n-1}\subset\mathbb R^{2n}$ is constructed that is discretely but not isotopically realizable. It is shown that for $n\equiv0,\,1\pmod4$ any map $f\colon S^n\to\mathbb R^{2n-2}\subset\mathbb R^{2n}$ is isotopically realizable, and for $n\equiv2\pmod4$, so also is every map $f\colon S^n\to\mathbb R^{2n-3}\subset\mathbb R^{2n}$. If $n\geqslant13$ and $n+1$ is not a power of $2$, an arbitrary map $f\colon S^n\to\mathbb R^{5[n/3]+3}\subset\mathbb R^{2n}$ is isotopically realizable.
The main results are devoted to the isotopic realization problem for maps $f$ of the form $S^n\stackrel{f}\to S^n\subset\mathbb R^{2n}$, $n=2^l-1$. It is established that if it has a negative solution, then the inverse images of points under the map $f$ have a certain homology property connected with actions of the group of $p$-adic integers. The solution is affirmative if $f$ is Lipschitzian and its van Kampen–Skopenkov thread has finite order. In connection with the proof the functors $\operatorname{Ext}_{\square}$ and $\operatorname{Ext}_{\bowtie}$ in the relative homology algebra of inverse spectra are introduced.