The heegaard distances cover all nonnegative integers

We prove two main results: (1) For any integers n ≥ 1 and g ≥ 2, there is a closed 3-manifold Mⁿ_g admitting a distance-n, genus-g Heegaard splitting, unless (g, n) = (2, 1). Furthermore, Mⁿ_g can be chosen to be hyperbolic unless (g, n) = (3, 1). (2) For any integers g ≥ 2 and n ≥ 4, there are infinitely many nonhomeomorphic closed 3-manifolds admitting distance-n, genus-g Heegaard splittings.