Local linking

A property of the disposition of a closed set $\Phi$ close to a point $a$ of it in a Euclidean space $\mathbf R^n$. It consists of the existence of a number $\epsilon>0$ such that, for any positive number $\delta$, in the open set $O(a,\delta)\setminus\Phi$ there lies a $q$-dimensional cycle $Z^q$, $q<n$, with integer coefficients, having the following property: Any compact set $P$ lying in $O(a,\epsilon)$ in which $Z^q$ is homologous to zero has non-empty intersection with $\Phi$. Here $O(a,\delta)$ and $O(a,\epsilon)$ are spheres with centre $a$ and radii $\delta$ and $\epsilon$. Without changing the content of this definition one can restrict oneself to compact sets $P$ that are polyhedra. For $q=0$ 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 $\dim\Phi=p$ it is necessary and sufficient that the number $n-p-1$ should be the smallest integer $q$ for which there is a $q$-dimensional linking of $\Phi$ in $\mathbf R^n$ close to some point $a\in\Phi$. An analogous theorem has been proved concerning obstructions "modulo m" , which characterizes sets $\Phi$ that have homological dimension $p$ "modulo m" .

Far-reaching generalizations of obstruction theorems are theorems on the homological containment of compact sets.

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