I am looking for a term to describe manifolds that are connected but not simply connected. Multi-connected looks like a strong candidate. However, I can't seem to find a formal definition of the concept. What is the precise definition of a multi-connected manifold?
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2You already gave it: a manifold that's connected but not simply connected. I think this is old language, though. – Qiaochu Yuan Dec 17 '18 at 00:07
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How would you term it? – Asdf Dec 17 '18 at 00:14
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2Just "connected but not simply connected." It's not a concept I have much of a need for so I'm okay not having a term for it. – Qiaochu Yuan Dec 17 '18 at 00:15
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Alright. I'll take multi-connected. – Asdf Dec 17 '18 at 00:18
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A manifold that is connected (so path-connected too) but not simply connected, i.e. $\pi_1(X,x_0)$ is not trivial (for some choice of base point, it matters not which, by path-connectedness).
Henno Brandsma
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I am not convinced that the expression "multi-connected" is a good choice for a connected but non-simply connected manifold.
The expression "doubly connected" is frequently used in complex analysis. A doubly connected region in the complex plane is a region bounded by two Jordan curves. See for example Conformal map of doubly connected domain into annulus..
A generalization is "multiply connected". See https://www.encyclopediaofmath.org/index.php/Multiply-connected_domain. Here are some references.
Walsh, J. L. "On the conformal mapping of multiply connected regions." Transactions of the American Mathematical Society 82.1 (1956): 128-146.
Walsh, J. L., and H. J. Landau. "On canonical conformal maps of multiply connected regions." Transactions of the American Mathematical Society 93.1 (1959): 81-96.
Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.
Paul Frost
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Multiply connected looks like well-established terminology. It has a Mathworld page http://mathworld.wolfram.com/MultiplyConnected.html I found multi-connected in some physics literature, but it bothered me that it did not seem to turn up in the mathematics literature. – Asdf Dec 18 '18 at 05:33