In this paper we study the isotopic realization problem, which is the question of isotopic realizability of a given (continuous) map , that is, the possibility of a uniform approximation of by a continuous family of embeddings , , under the condition that is discretely realizable, that is, that there exists a uniform approximation of by a sequence of embeddings , . For each a map 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 a map is constructed that is discretely but not isotopically realizable. It is shown that for any map is isotopically realizable, and for , so also is every map . If and is not a power of 2, an arbitrary map is isotopically realizable. The main results are devoted to the isotopic realization problem for maps of the form , . It is established that if it has a negative solution, then the inverse images of points under the map have a certain homology property connected with actions of the group of -adic integers. The solution is affirmative if is Lipschitzian and its van Kampen-Skopenkov thread has finite order. In connection with the proof the functors and in the relative homology algebra of inverse spectra are introduced.