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This article is cited in 1 scientific paper (total in 1 paper)
On isotopic realizability of maps factored through a hyperplane
S. A. Melikhov Steklov Mathematical Institute, Russian Academy of Sciences
Abstract:
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 $g_t$, $t\in[0,\infty)$, under the condition that $f$ is discretely realizable, that is,
that there exists a uniform approximation of $f$ by a sequence of embeddings $h_n$, $n\in\mathbb N$.
For each $n\geqslant3$ a map $f\colon S^n\to\mathbb R^{2n}$ 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 $n\geqslant4$ a map
$f\colon S^n\to\mathbb R^{2n-1}\subset\mathbb R^{2n}$
is constructed that is discretely but not isotopically realizable. It is shown that for
$n\equiv0, 1\pmod4$ any map
$f\colon S^n\to\mathbb R^{2n-2}\subset\mathbb R^{2n}$
is isotopically realizable, and for $n\equiv2\pmod4$, so also is every map
$f\colon S^n\to\mathbb R^{2n-3}\subset\mathbb R^{2n}$.
If $n\geqslant13$ and $n+1$ is not a power of $2$,
an arbitrary map
$f\colon S^n\to\mathbb R^{5[n/3]+3}\subset\mathbb R^{2n}$ is isotopically realizable.
The main results are devoted to the isotopic realization
problem for maps $f$ of the form
$S^n\stackrel{f}\to S^n\subset\mathbb R^{2n}$, $n=2^l-1$.
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 $\operatorname{Ext}_{\square}$ and $\operatorname{Ext}_{\bowtie}$
in the relative homology algebra of inverse spectra are introduced.
DOI:
https://doi.org/10.4213/sm839
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English version:
Sbornik: Mathematics, 2004, 195:8, 1117–1163
Bibliographic databases:
UDC:
515.1
MSC: Primary 57Q35; Secondary 55N07, 55N22, 57Q15, 57Q37, 57Q45, 57Q91, 55S20, 5 Received: 26.08.2002 and 12.01.2004
Citation:
S. A. Melikhov, “On isotopic realizability of maps factored through a hyperplane”, Mat. Sb., 195:8 (2004), 47–90; Sb. Math., 195:8 (2004), 1117–1163
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/msb839https://doi.org/10.4213/sm839 http://mi.mathnet.ru/eng/msb/v195/i8/p47
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This publication is cited in the following articles:
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Melikhov S.A., Repovš D., “$n$-quasi-isotopy. I. Questions of nilpotence”, J. Knot Theory Ramifications, 14:5 (2005), 571–602
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