The Heegaard distances cover all non-negative integers

الأصل · EN

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