Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Representation of the Reeb Graph as a Sub-Complex of Manifold   The Reeb graph $\mathcal{R}(f) $ is one of the fundamental invariants of a
smooth function $f\colon M\to \mathbb{R} $ with isolated critical points. It is
defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a
relation that depends on $f$. Here we construct a $1$-dimensional complex
$\Gamma(f)$ embedded into $M$ which is homotopy equivalent to $\mathcal{R}(f)$.
As a consequence we show that for every function $f$ on a manifold with finite
fundamental group, the Reeb graph of $f$ is a tree. If $\pi_1(M)$ is an abelian
group, or more general, a discrete amenable group, then $\mathcal{R}(f)$
contains at most one loop. Finally we prove that the number of loops in the
Reeb graph of every function on a surface $M_g$ is estimated from above by $g$,
the genus of $M_g$.

1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an