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

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
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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
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\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
\mathnet{http://mi.mathnet.ru/tm1880}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2603266}
\zmath{https://zbmath.org/?q=an:1196.57019}
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\jour Proc. Steklov Inst. Math.
\yr 2009
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\crossref{https://doi.org/10.1134/S0081543809030092}
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3. Freedman M., Krushkal V., “Geometric Complexity of Embeddings in R-D”, Geom. Funct. Anal., 24:5 (2014), 1406–1430
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