What are the Jordan curve theorems for higher genus surfaces? I'm thinking of something like a surface with n handles can be represented as a 2(n+1) sided polygon Q with opposite sides identified. Let C be a "closed" non-self-intersecting curve from a point p on one side to the "point on the opposite side" identified with p. Then Q \ C is `` topologically equivalent" to a 2k sided polygon with opposite sides identified where k<n+1.
2026-03-28 03:52:59.1774669979
What are the Jordan curve theorems for higher genus surfaces?
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Your question is ambiguous, but here are some examples of what's true:
Suppose that $S$ is a (possibly non-orientable) connected surface (without boundary) and $\alpha\subset S$ is a subset homeomorphic to $S^1$ (a Jordan curve). Then $S'=S\setminus \alpha$ is either connected or consists of exactly two components.
If, additionally, one assumes that $S$ is compact, then one can also prove that there are only finitely many homeomorphism types of surfaces $S'$ above. If $\alpha$ is non-separating then there is precisely one homeomorphism type of $S'$. (The possible types depend on the topology of $S$.)
Proofs of such results are long. Consider reading for instance
Farb, Benson; Margalit, Dan, A primer on mapping class groups, Princeton Mathematical Series. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/ebook). xiv, 492 p. (2011). ZBL1245.57002.