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 Zap. Nauchn. Sem. POMI, 2000, Volume 267, Pages 53–87 (Mi znsl1266)

On isotopic realizability of continuous mappings

P. M. Akhmet'eva, S. A. Melikhovb

a Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation
b M. V. Lomonosov Moscow State University

Abstract: Under the metastable dimension restriction, we present an algebraic description of the class of discretely realizable maps (i.e., maps that are arbitrarily closely approximable by embeddings) which fail to be isotopically realizable (i.e., to be obtained from an embedding via a pseudo-isotopy). These maps are precisely the ones which yield a negative solution to the Isotopic Realization Problem of E. V. Shchepin (1993), see [1, 27].
A cohomological obstruction for isotopic realizability of a discretely realizable map of an $n$-polyhedron into an orientable PL $m$-manifold is constructed. We also present an obstruction for discrete realizability of a map $S^n\to\mathbb R^m$. If $m>\frac{3(n+1)}2$, these obstructions are shown to be complete. In fact, the latter obstruction can be regarded as an element of the limit of certain inverse spectrum of finitely generated Abelian groups (which are cohomology groups of compact polyhedra with coefficients in a locally constant sheaf), while the first obstruction can be identified with an element of the derived limit of this spectrum. On the other hand, the obstructions generalize the classical van Kampen obstruction for embeddability of an $n$-polyhedron into $\mathbb R^{2n}$.
An explicit construction of a series of discretely but not isotopically realizable maps $S^n\to\mathbb R^{2n}$ is given for $n\geqslant3$. The singular sets of these maps are homeomorphic to the disjoint union of the $p$-adic solenoid, $p\geqslant3$, and a point. Furthermore, it is shown that the Isotopic Realization Problem has positive solution in the metastable range under the assumption of stabilization with codimension 1, or if the configuration singular set $\Sigma(f)=\{(x,y)\in S^n\times S^n\mid f(x)=f(y)\}$ of a map $f\colon S^n\to\mathbb R^m$ is acyclic in dimension $2n-m$ with respect to the Steenrod–Sitnikov homology.

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English version:
Journal of Mathematical Sciences (New York), 2003, 113:6, 759–776

Bibliographic databases:

UDC: 515.164.6+515.163.6+515.126.2

Citation: P. M. Akhmet'ev, S. A. Melikhov, “On isotopic realizability of continuous mappings”, Geometry and topology. Part 5, Zap. Nauchn. Sem. POMI, 267, POMI, St. Petersburg, 2000, 53–87; J. Math. Sci. (N. Y.), 113:6 (2003), 759–776

Citation in format AMSBIB
\Bibitem{AkhMel00} \by P.~M.~Akhmet'ev, S.~A.~Melikhov \paper On isotopic realizability of continuous mappings \inbook Geometry and topology. Part~5 \serial Zap. Nauchn. Sem. POMI \yr 2000 \vol 267 \pages 53--87 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl1266} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1809817} \zmath{https://zbmath.org/?q=an:1039.57014} \transl \jour J. Math. Sci. (N. Y.) \yr 2003 \vol 113 \issue 6 \pages 759--776 \crossref{https://doi.org/10.1023/A:1021275032646} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. A. Melikhov, R. V. Mikhailov, “Links modulo knots and the isotopic realization problem”, Russian Math. Surveys, 56:2 (2001), 414–415
2. Melikhov S.A., “On maps with unstable singularities”, Topology and Its Applications, 120:1–2 (2002), 105–156
3. S. A. Melikhov, “Isotopic and continuous realizability of maps in the metastable range”, Sb. Math., 195:7 (2004), 983–1016
4. S. A. Melikhov, “On isotopic realizability of maps factored through a hyperplane”, Sb. Math., 195:8 (2004), 1117–1163
5. P. M. Akhmet'ev, “A Remark on the Realization of Mappings of the 3-Dimensional Sphere into Itself”, Proc. Steklov Inst. Math., 247 (2004), 4–8
6. S. A. Melikhov, “Sphere Eversions and Realization of Mappings”, Proc. Steklov Inst. Math., 247 (2004), 143–163
7. Akhmetiev P.M., “Pontryagin–Thom construction for approximation of mappings by embeddings”, Topology and Its Applications, 140:2–3 (2004), 133–149