Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Heegaard distances cover all non-negative integers   In this paper, we prove that
  (1) For any integers $n\geq 1$ and $g\geq 2$, 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 $g\geq 2$ and $n\geq 4$, there are infinitely many
non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings
of genus $g$.

26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio