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)