Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A characteristics for a surface sum of two handlebodies along an annulus
  or a once-punctured torus to be a handlebody   The main results of the paper is that we give a characteristics for an
annulus sum and a once-punctured torus sum of two handlebodies to be a
handlebody as follows: 1. The annulus sum $H=H_1\cup_A H_2$ of two handlebodies
$H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a
longitude for either $H_1$ or $H_2$. 2. Let $H=H_1\cup_T H_2$ be a surface sum
of two handlebodies $H_1$ and $H_2$ along a once-punctured torus $T$. Suppose
that $T$ is incompressible in both $H_1$ and $H_2$. Then $H$ is a handlebody if
and only if the there exists a collection $\{\delta, \sigma\}$ of simple closed
curves on $T$ such that either $\{\delta, \sigma\}$ is primitive in $H_1$ or
$H_2$, or $\{\delta\}$ is primitive in $H_1$ and $\{\sigma\}$ is primitive in
$H_2$.

3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess