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Tr. Mat. Inst. Steklova, 2009, Volume 266, Pages 149–183 (Mi tm1880)  

This article is cited in 7 scientific papers (total in 7 papers)

The van Kampen Obstruction and Its Relatives

S. A. Melikhov

Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

Abstract: We review a cochain-free treatment of the classical van Kampen obstruction $\vartheta$ to embeddability of an $n$-polyhedron in $\mathbb R^{2n}$ and consider several analogs and generalizations of $\vartheta$, including an extraordinary lift of $\vartheta$, which has been studied by J.-P. Dax in the manifold case. The following results are obtained:
(1) The $\mod2$ reduction of $\vartheta$ is incomplete, which answers a question of Sarkaria.
(2) An odd-dimensional analog of $\vartheta$ is a complete obstruction to linkless embeddability ($= $“intrinsic unlinking”) of a given $n$-polyhedron in $\mathbb R^{2n+1}$.
(3) A “blown-up” one-parameter version of $\vartheta$ is a universal type 1 invariant of singular knots, i.e., knots in $\mathbb R^3$ with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram ($= $Polyak–Viro) formula.
(4) Settling a problem of Yashchenko in the metastable range, we find that every PL manifold $N$ nonembeddable in a given $\mathbb R^m$, $m\ge\frac{3(n+1)}2$, contains a subset $X$ such that no map $N\to\mathbb R^m$ sends $X$ and $N\setminus X$ to disjoint sets.
(5) We elaborate on McCrory's analysis of the Zeeman spectral sequence to geometrically characterize “$k$-co-connected and locally $k$-co-connected” polyhedra, which we embed in $\mathbb R^{2n-k}$ for $k<\frac{n-3}2$, thus extending the Penrose–Whitehead–Zeeman theorem.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2009, 266, 142–176

Bibliographic databases:

UDC: 515.164.6+515.162.8+515.148
Received in May 2009

Citation: S. A. Melikhov, “The van Kampen Obstruction and Its Relatives”, Geometry, topology, and mathematical physics. II, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday, Tr. Mat. Inst. Steklova, 266, MAIK Nauka/Interperiodica, Moscow, 2009, 149–183; Proc. Steklov Inst. Math., 266 (2009), 142–176

Citation in format AMSBIB
\by S.~A.~Melikhov
\paper The van Kampen Obstruction and Its Relatives
\inbook Geometry, topology, and mathematical physics.~II
\bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2009
\vol 266
\pages 149--183
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2009
\vol 266
\pages 142--176

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    This publication is cited in the following articles:
    1. Matoušek J., Tancer M., Wagner U., “Hardness of embedding simplicial complexes in $\mathbb R^d$”, J. Eur. Math. Soc. (JEMS), 13:2 (2011), 259–295  crossref  mathscinet  zmath  isi  scopus
    2. Wagner U., “Minors in random and expanding hypergraphs”, Computational Geometry (SCG 11), 2011, 351–360  mathscinet  zmath  isi
    3. Freedman M., Krushkal V., “Geometric Complexity of Embeddings in R-D”, Geom. Funct. Anal., 24:5 (2014), 1406–1430  crossref  mathscinet  zmath  isi  elib  scopus
    4. Goncalves D., Skopenkov A., “a Useful Lemma on Equivariant Maps”, Homol. Homotopy Appl., 16:2 (2014), 307–309  crossref  mathscinet  zmath  isi  elib  scopus
    5. S. A. Melikhov, “Transverse fundamental group and projected embeddings”, Proc. Steklov Inst. Math., 290:1 (2015), 155–165  mathnet  crossref  crossref  isi  elib  elib
    6. Oleg R. Musin, Alexey Yu. Volovikov, “Borsuk–Ulam type spaces”, Mosc. Math. J., 15:4 (2015), 749–766  mathnet  crossref  mathscinet
    7. A. B. Skopenkov, “A user's guide to the topological Tverberg conjecture”, Russian Math. Surveys, 73:2 (2018), 323–353  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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