The Heegaard distances cover all non-negative integers

Abstract

In this paper, we prove that (1) For any integers $n\ge 1$ and $g\ge 2$, there is a closed 3-manifold $M_g^n$ which admits a distance $n$ Heegaard splitting of genus $g$ except that the pair of $(g,n)$ is $(2,1)$. Furthermore, $M_g^n$ can be chosen to be hyperbolic except that the pair of $(g,n)$ is $(3,1)$. (2) For any integers $g\ge 2$ and $n\ge 4$, there are infinitely many non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings of genus $g$.

20 pages, 8 figures 2 References Added, title changed

References in corpus (9)

• Geometry of the Complex of Curves I: Hyperbolicity
• Alternate Heegaard genus bounds distance
• High distance knots
• Sweepouts of amalgamated 3-manifolds
• Heegaard splittings of distance exactly $n$
• Heegaard surfaces and the distance of amalgamation
• Horizontal Dehn Surgery and genericity in the curve complex
• Bounds on exceptional Dehn filling II
• Small 3-manifolds with large Heegaard distance