A necessary and sufficient condition for a 3-manifold to have genus \(g\) Heegaard splitting (a proof of Hass-Thompson conjecture)
The condition referred to in the title is that there exist a genus \(g\) handlebody in the manifold such that any knot in the manifold can be ambient isotoped to lie inside the handlebody. The genus 0 case was obtained by \textit{R. H. Bing} [Ann. Math., II. Ser. 68, 17-37 (1958; Zbl 0081.392)] and the genus 1 case by \textit{J. Hass} and \textit{A. Thompson} [Proc. Am. Math. Soc. 107, No. 4, 1107-1110 (1989; Zbl 0694.57006)].
scientific article; zbMATH DE number 639567