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