Multiply-connected domain

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