Local linking
A property of the disposition of a closed set close to a point
of it in a Euclidean space
. It consists of the existence of a number
such that, for any positive number
, in the open set
there lies a
-dimensional cycle
,
, with integer coefficients, having the following property: Any compact set
lying in
in which
is homologous to zero has non-empty intersection with
. Here
and
are spheres with centre
and radii
and
. Without changing the content of this definition one can restrict oneself to compact sets
that are polyhedra. For
the concept of a local linking goes over to the concept of a local cut (cf. Local decomposition). Aleksandrov's obstruction theorem: In order that
it is necessary and sufficient that the number
should be the smallest integer
for which there is a
-dimensional linking of
in
close to some point
. An analogous theorem has been proved concerning obstructions "modulo m" , which characterizes sets
that have homological dimension
"modulo m" .
Far-reaching generalizations of obstruction theorems are theorems on the homological containment of compact sets.
References
[1] | P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian) |
[2] | K. Sitnikov, "On homological girdling of compacta in Euclidean space" Dokl. Akad. Nauk SSSR , 81 (1951) pp. 153–156 (In Russian) |
Local linking. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_linking&oldid=43441