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This article is cited in 4 scientific papers (total in 4 papers)
Isotopic and continuous realizability of maps in the metastable range
S. A. Melikhovab a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Florida
Abstract:
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.
Received: 26.08.2002 and 12.01.2004
Citation:
S. A. Melikhov, “Isotopic and continuous realizability of maps in the metastable range”, Sb. Math., 195:7 (2004), 983–1016
Linking options:
https://www.mathnet.ru/eng/sm835https://doi.org/10.1070/SM2004v195n07ABEH000835 https://www.mathnet.ru/eng/sm/v195/i7/p71
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