We show that the homeomorphism problem for arbitrary (noncompact) surfaces is complete for countable structures in the sense of Borel complexity, meaning it is of a maximum complexity among all analytic equivalence relations Borel reducible to the isomorphism problem on some class of countable first-order structures. To do so, we formulate and prove Borel measurable forms of the classical Jordan—Schoenflies and triangulation theorems for surfaces.
This work is joint with Jeffrey Bergfalk. (This talk will be a sequel, with more details, to the Logic Colloquium from Oct 23, 2025.)