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 Mat. Sb., 2004, Volume 195, Number 7, Pages 71–104 (Mi msb835)

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.

DOI: https://doi.org/10.4213/sm835

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English version:
Sbornik: Mathematics, 2004, 195:7, 983–1016

Bibliographic databases:

UDC: 515.1
MSC: Primary 57Q35; Secondary 55N07, 55N22, 57Q15, 57Q37, 57Q45, 57Q91, 55S20, 5

Citation: S. A. Melikhov, “Isotopic and continuous realizability of maps in the metastable range”, Mat. Sb., 195:7 (2004), 71–104; Sb. Math., 195:7 (2004), 983–1016

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb835
• https://doi.org/10.4213/sm835
• http://mi.mathnet.ru/eng/msb/v195/i7/p71

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This publication is cited in the following articles:
1. S. A. Melikhov, “On isotopic realizability of maps factored through a hyperplane”, Sb. Math., 195:8 (2004), 1117–1163
2. Melikhov S.A., Repovš D., “$n$-quasi-isotopy. I. Questions of nilpotence”, J. Knot Theory Ramifications, 14:5 (2005), 571–602
3. S. A. Melikhov, “Steenrod homotopy”, Russian Math. Surveys, 64:3 (2009), 469–551
4. Proc. Steklov Inst. Math., 266 (2009), 142–176
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