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''in a path-connected space''
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A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655501.png" /> in which there are closed paths not homotopic to zero, or, in other words, whose [[Fundamental group|fundamental group]] is not trivial. This means that there are closed paths in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655502.png" /> which cannot be continuously deformed to a point while remaining throughout within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655503.png" />, or, otherwise, a multiply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655504.png" /> is a domain which is not a [[Simply-connected domain|simply-connected domain]].
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The order of connectivity of a plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655505.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655506.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655507.png" /> (or in the compactification of these spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655508.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m0655509.png" />) is the number of (homologically) independent one-dimensional cycles, that is, the one-dimensional [[Betti number|Betti number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555011.png" />. If the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555012.png" /> of connected components of the boundary of a plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555013.png" />, considered as a domain in the compactified space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555015.png" />, is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555016.png" />; otherwise one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555017.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555019.png" /> is a simply-connected domain, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555020.png" /> it is a finitely-connected domain (one also uses such terms as doubly-connected domain, triply-connected domain<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555021.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555023.png" />-connected domain), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555025.png" /> is an infinitely-connected domain. All plane finitely-connected domains with equal order of connectivity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555026.png" />, are homeomorphic to each other. By removing from such a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555027.png" /> all the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555028.png" /> cuts, that is, Jordan arcs joining pairs of connected components of the boundary, it is always possible to obtain a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555029.png" />. About the conformal types of plane multiply-connected domains see [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]].
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''in a path-connected space''
  
The topological types of domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555031.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555033.png" />, are far more diverse and cannot be characterized by a single number. Here, sometimes, the term  "multiply-connected domain" (with various provisos) is used when the fundamental group is trivial but some higher-dimensional [[Homology group|homology group]] is not trivial.
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A domain  $  D $
 +
in which there are closed paths not homotopic to zero, or, in other words, whose [[Fundamental group|fundamental group]] is not trivial. This means that there are closed paths in  $  D $
 +
which cannot be continuously deformed to a point while remaining throughout within  $  D $,
 +
or, otherwise, a multiply-connected domain  $  D $
 +
is a domain which is not a [[Simply-connected domain|simply-connected domain]].
  
 +
The order of connectivity of a plane domain  $  D $
 +
in  $  \mathbf R  ^ {2} $
 +
or  $  \mathbf C = \mathbf C  ^ {1} $(
 +
or in the compactification of these spaces,  $  \overline{\mathbf R}\; {}  ^ {2} $
 +
or  $  \overline{\mathbf C}\; $)
 +
is the number of (homologically) independent one-dimensional cycles, that is, the one-dimensional [[Betti number|Betti number]]  $  p  ^ {1} $
 +
of  $  D $.
 +
If the number  $  k $
 +
of connected components of the boundary of a plane domain  $  D $,
 +
considered as a domain in the compactified space  $  \overline{\mathbf R}\; {}  ^ {2} $
 +
or  $  \overline{\mathbf C}\; $,
 +
is finite, then  $  p  ^ {1} = k $;
 +
otherwise one sets  $  p  ^ {1} = \infty $.
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When  $  p  ^ {1} = 1 $,
 +
$  D $
 +
is a simply-connected domain, when  $  p  ^ {1} < \infty $
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it is a finitely-connected domain (one also uses such terms as doubly-connected domain, triply-connected domain $  \dots $
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$  k $-
 +
connected domain), when  $  p  ^ {1} = \infty $,
 +
$  D $
 +
is an infinitely-connected domain. All plane finitely-connected domains with equal order of connectivity,  $  k $,
 +
are homeomorphic to each other. By removing from such a domain  $  D $
 +
all the points of  $  k - 1 $
 +
cuts, that is, Jordan arcs joining pairs of connected components of the boundary, it is always possible to obtain a simply-connected domain  $  D  ^ {*} \subset  D $.
 +
About the conformal types of plane multiply-connected domains see [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]].
  
 +
The topological types of domains in  $  \mathbf R  ^ {n} $,
 +
$  n \geq  3 $,
 +
or  $  \mathbf C  ^ {m} $,
 +
$  m \geq  2 $,
 +
are far more diverse and cannot be characterized by a single number. Here, sometimes, the term  "multiply-connected domain"  (with various provisos) is used when the fundamental group is trivial but some higher-dimensional [[Homology group|homology group]] is not trivial.
  
 
====Comments====
 
====Comments====
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There are two rather different concepts which go by the phrase  "multi-connected spacemulti-connected"  or  "multiply-connected" . The concept and terminology as described above come from the theory of functions of a complex variable.
 
There are two rather different concepts which go by the phrase  "multi-connected spacemulti-connected"  or  "multiply-connected" . The concept and terminology as described above come from the theory of functions of a complex variable.
  
On the other hand, in (algebraic) topology one defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555035.png" />-connected space as a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555036.png" /> such that any mapping from a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555038.png" />, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555039.png" /> is homotopic to zero. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065550/m06555040.png" />-connectedness is the same as path connectedness.
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On the other hand, in (algebraic) topology one defines an $  n $-
 +
connected space as a space $  X $
 +
such that any mapping from a sphere $  S  ^ {m} $,  
 +
m \leq  n $,  
 +
into $  X $
 +
is homotopic to zero. Thus, 0 $-
 +
connectedness is the same as path connectedness.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.K. Francis,  "A topological picturebook" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.S. Massey,  "Algebraic topology: an introduction" , Springer  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. 23; 415–455</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.K. Francis,  "A topological picturebook" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.S. Massey,  "Algebraic topology: an introduction" , Springer  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)  pp. 23; 415–455</TD></TR></table>

Latest revision as of 08:02, 6 June 2020


in a path-connected space

A domain $ D $ in which there are closed paths not homotopic to zero, or, in other words, whose fundamental group is not trivial. This means that there are closed paths in $ D $ which cannot be continuously deformed to a point while remaining throughout within $ D $, or, otherwise, a multiply-connected domain $ D $ is a domain which is not a simply-connected domain.

The order of connectivity of a plane domain $ D $ in $ \mathbf R ^ {2} $ or $ \mathbf C = \mathbf C ^ {1} $( or in the compactification of these spaces, $ \overline{\mathbf R}\; {} ^ {2} $ or $ \overline{\mathbf C}\; $) is the number of (homologically) independent one-dimensional cycles, that is, the one-dimensional Betti number $ p ^ {1} $ of $ D $. If the number $ k $ of connected components of the boundary of a plane domain $ D $, considered as a domain in the compactified space $ \overline{\mathbf R}\; {} ^ {2} $ or $ \overline{\mathbf C}\; $, is finite, then $ p ^ {1} = k $; otherwise one sets $ p ^ {1} = \infty $. When $ p ^ {1} = 1 $, $ D $ is a simply-connected domain, when $ p ^ {1} < \infty $ it is a finitely-connected domain (one also uses such terms as doubly-connected domain, triply-connected domain $ \dots $ $ k $- connected domain), when $ p ^ {1} = \infty $, $ D $ is an infinitely-connected domain. All plane finitely-connected domains with equal order of connectivity, $ k $, are homeomorphic to each other. By removing from such a domain $ D $ all the points of $ k - 1 $ cuts, that is, Jordan arcs joining pairs of connected components of the boundary, it is always possible to obtain a simply-connected domain $ D ^ {*} \subset D $. About the conformal types of plane multiply-connected domains see Riemann surfaces, conformal classes of.

The topological types of domains in $ \mathbf R ^ {n} $, $ n \geq 3 $, or $ \mathbf C ^ {m} $, $ m \geq 2 $, are far more diverse and cannot be characterized by a single number. Here, sometimes, the term "multiply-connected domain" (with various provisos) is used when the fundamental group is trivial but some higher-dimensional homology group is not trivial.

Comments

For a discussion of non-planar multiply-connected domains see [a1].

There are two rather different concepts which go by the phrase "multi-connected spacemulti-connected" or "multiply-connected" . The concept and terminology as described above come from the theory of functions of a complex variable.

On the other hand, in (algebraic) topology one defines an $ n $- connected space as a space $ X $ such that any mapping from a sphere $ S ^ {m} $, $ m \leq n $, into $ X $ is homotopic to zero. Thus, $ 0 $- connectedness is the same as path connectedness.

References

[a1] G.K. Francis, "A topological picturebook" , Springer (1987)
[a2] W.S. Massey, "Algebraic topology: an introduction" , Springer (1967)
[a3] G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 23; 415–455
How to Cite This Entry:
Multiply-connected domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiply-connected_domain&oldid=47941
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article