Isotopic and continuous realizability of maps in the metastable range

S. A. Melikhov

Sbornik. Mathematics, Tome 195 (2004) no. 7, pp. 983-1016

Résumé

A continuous map $f$ of a compact $n$-polyhedron into an orientable piecewise linear $m$-manifold, $m-n\geqslant3$, is discretely (isotopically) realizable if it is the uniform limit of a sequence of embeddings $g_k$, $k\in\mathbb N$ (respectively, of an isotopy $g_t$, $t\in[0,\infty)$), and is continuously realizable if any embedding sufficiently close to $f$ can be included in an arbitrarily small such isotopy. It was shown by the author that for $m=2n+1$, $n\ne1$, all maps are continuously realizable, but for $m=3$, $n=6$ there are maps that are discretely realizable, but not isotopically. The first obstruction $o(f)$ to the isotopic realizability of a discretely realizable map $f$ lies in the kernel $K_f$ of the canonical epimorphism between the Steenrod and Čech $(2n-m)$-dimensional homologies of the singular set of $f$. It is known that for $m=2n$, $n\geqslant4$, this obstruction is complete and $f$ is continuously realizable if and only if the group $K_f$ is trivial. In the present paper it is established that $f$ is continuously realizable if and only if $K_f$ is trivial even in the metastable range, that is, for $m\geqslant3(n+1)/2$, $n\ne1$. The proof uses higher cohomology operations. On the other hand, for each $n\geqslant9$ a map $S^n\to\mathbb R^{2n-5}$ is constructed that is discretely realizable and has zero obstruction $o(f)$ to the isotopic realizability, but is not isotopically realizable, which fact is detected by the Steenrod square. Thus, in order to determine whether a discretely realizable map in the metastable range is isotopically realizable one cannot avoid using the complete obstruction in the group of Koschorke–Akhmet'ev bordisms.