What is the precise definition of a multi-connected manifold? 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?
 A: 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).
A: 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.
https://www.ams.org/journals/tran/1956-082-01/S0002-9947-1956-0080727-2/S0002-9947-1956-0080727-2.pdf
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.
https://www.ams.org/journals/tran/1959-093-01/S0002-9947-1959-0160884-2/S0002-9947-1959-0160884-2.pdf
Landau, H. J. "On canonical conformal maps of multiply connected domains." Transactions of the American Mathematical Society 99.1 (1961): 1-20.
https://www.ams.org/journals/tran/1961-099-01/S0002-9947-1961-0121474-X/S0002-9947-1961-0121474-X.pdf
