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 gt, t element of [0,∞), under the condition that f is discretely realizable, that is, that there exists a uniform approximation of f by a sequence of embeddings hn, n element of N. For each n≥3 a map f:Sn→R2n 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≥4 a map f:Sn→R2n-1 subset of R2n is constructed that is discretely but not isotopically realizable. It is shown that for n≡0,1 (mod 4) any map f:Sn→R2n-2 subset of R2n is isotopically realizable, and for n≡2 (mod 4), so also is every map f:Sn→R2n-3 subset of R2n. If n≥13 and n+1 is not a power of 2, an arbitrary map f:Sn→R5[n/3]+3 subset of R2n is isotopically realizable. The main results are devoted to the isotopic realization problem for maps f of the form Sn→fSn subset of R2n, n=2l-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 Ext-square and Ext-bowtie in the relative homology algebra of inverse spectra are introduced.