A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical

Abstract

Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M. In this article, we prove that either the disk complex D(F) is contractible or F is critical. Hence, the topological index of F is two if F is topologically minimal.

20 pages, 11 figures. More detailed arguments and figures

References in corpus (3)

• Heegaard splittings of distance exactly $n$
• On critical Heegaard splittings of tunnel number two composite knot exteriors
• Critical Heegaard surfaces obtained by self-amalgamation

Cited by in corpus (1)

• On the group action of $Mod(M,F)$ on the disk complex