Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Round fold maps on $3$--manifolds   We show that a closed orientable 3--dimensional manifold admits a round fold
map into the plane, i.e. a fold map whose critical value set consists of
disjoint simple closed curves isotopic to concentric circles, if and only if it
is a graph manifold, generalizing the characterization for simple stable maps
into the plane. Furthermore, we also give a characterization of closed
orientable graph manifolds that admit directed round fold maps into the plane,
i.e.\ round fold maps such that the number of regular fiber components of a
regular value increases toward the central region in the plane.

8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph