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A property of the disposition of a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601701.png" /> close to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601702.png" /> of it in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601703.png" />. It consists of the existence of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601704.png" /> such that, for any positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601705.png" />, in the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601706.png" /> there lies a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601707.png" />-dimensional cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l0601709.png" />, with integer coefficients, having the following property: Any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017010.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017011.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017012.png" /> is homologous to zero has non-empty intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017013.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017015.png" /> are spheres with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017016.png" /> and radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017018.png" />. Without changing the content of this definition one can restrict oneself to compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017019.png" /> that are polyhedra. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017020.png" /> the concept of a local linking goes over to the concept of a local cut (cf. [[Local decomposition|Local decomposition]]). Aleksandrov's obstruction theorem: In order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017021.png" /> it is necessary and sufficient that the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017022.png" /> should be the smallest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017023.png" /> for which there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017024.png" />-dimensional linking of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017026.png" /> close to some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017027.png" />. An analogous theorem has been proved concerning obstructions  "modulo m" , which characterizes sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017028.png" /> that have [[Homological dimension|homological dimension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060170/l06017029.png" /> "modulo m" .
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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|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|homological dimension]] $p$ "modulo m" .
  
 
Far-reaching generalizations of obstruction theorems are theorems on the [[Homological containment|homological containment]] of compact sets.
 
Far-reaching generalizations of obstruction theorems are theorems on the [[Homological containment|homological containment]] of compact sets.

Latest revision as of 11:41, 19 November 2018

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.

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)
How to Cite This Entry:
Local linking. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_linking&oldid=12956
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article