The van Kampen obstruction and its relatives |
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Authors: | Sergey A Melikhov |
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Institution: | 1.Steklov Mathematical Institute,Russian Academy of Sciences,Moscow,Russia |
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Abstract: | We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron in ℝ2n
and consider several analogs and generalizations of ϑ, including an extraordinary lift of ϑ, which has been studied by J.-P. Dax in the manifold case. The following results are obtained: (1) The mod 2 reduction of
ϑ is incomplete, which answers a question of Sarkaria. (2) An odd-dimensional analog of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of a given n-polyhedron in ℝ2n+1. (3) A “blown-up” one-parameter version of ϑ is a universal type 1 invariant of singular knots, i.e., knots in ℝ3 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 ℝ
m
, m ≥ $
\frac{{3(n + 1)}}
{2}
$
\frac{{3(n + 1)}}
{2}
, contains a subset X such that no map N → ℝ
m
sends X and N \ 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 ℝ2
n−k
for k < $
\frac{{n - 3}}
{2}
$
\frac{{n - 3}}
{2}
, thus extending the Penrose-Whitehead-Zeeman theorem. |
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Keywords: | |
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