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(Cross-posted from MSE.)

The paper Chang, S.-Y. A. and Yang, P. C.: Conformal deformation of metrics on S 2 . J. Differential Geom., 27(2), 1988 (DOI, MathSciNet) states in Proposition 2.3 that Moser's inequality holds for

a piecewise $C^2$, bounded, finitely connected domain in the plane with finite number of vertices.

(Edit: I replaced “piecewise“ by “piecewise $C^2$”.)

Unfortunately, the term “finitely connected“ is not defined in the paper and Google is of little help to me.

I also asked this question to a colleague who remembered running into the same issue and, being better at using Google than I am, finding multiple definitions which (at least on first sight) disagreed. In their case this was not too worrisome as they were studying convex domains which fulfill all these conditions. However, I would be interested what precisely is needed for the cited proposition.

Thus, I am asking for reference introducing this term and perhaps also for a discussion of different definitions. (Perhaps they all agree?)

Edit: Let me make my question more precise and ask whether the definitions given in the comments coincide: Is

  • fundamental group finitely generated

equivalent to

  • Betti number $p^1$ finite

(for piecewise $C^2$, bounded connected domains in the plane with finite number of vertices)?

Improve this question
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    $\begingroup$ If I had to guess, I would assume “finitely connected” means the fundamental group is finite, or finitely presented, or finitely generated. (cf “simply connected” and “multiply connected”) $\endgroup$ Commented May 3, 2022 at 13:07
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    $\begingroup$ If you give a list of competing definitions, the question would be easier to answer. I would guess that by a finitely-connected domain the authors mean a compact embedded surface in the plane whose boundary circles are polygonal curves. In other words, consider a plane disk $D$ with polygonal boundary and remove from the interior of $D$ the interiors of finitely many disjoint disks with polygonal boundaries. On the other hand, maybe they also permit removing finitely many points from the interior of $D$. $\endgroup$ Commented May 3, 2022 at 13:08
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    $\begingroup$ From the title of the book I infer that they are talking about a domain on a surface. For this case, "finitely connected" has the standard meaning that the fundamental group is finitely generated. $\endgroup$ Commented May 3, 2022 at 14:42
  • $\begingroup$ See encyclopediaofmath.org/wiki/Multiply-connected_domain $\endgroup$ Commented May 3, 2022 at 17:35
  • $\begingroup$ The question is actually not what "finitely-connected" means. It is what "piecewise, bounded, finitely connected domain in the plane with finite number of vertices" means. $\endgroup$ Commented May 3, 2022 at 21:21

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