Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Realizing a homology class of a compact manifold by a homology class of
  an explicit closed submanifold--a new approach to Thom's works on homology
  classes of submanifolds-   It is a classical important problem of differential topology by Thom; for a
homology class of a compact manifold, can we realize this by a closed
submanifold with no boundary? This is true if the degree of the class is
smaller or equal to the half of the dimension of the outer manifold under the
condition that the coefficient ring is Z_2. If the degree of the class is
smaller or equal to 6 or equal to k-2 or k-1 under the condition that the
coefficient ring is the integer ring where k is the dimension of the manifold,
then this is also true. As a specific study, for 4-dimensional closed
manifolds, the topologies (genera) of closed and connected surfaces realizing
given 2nd homology classes have been actively studied, for example.
  In the present paper, we consider the following similar problem; can we
realize a homology class of a compact manifold by a homology class of an
explicit closed manifold embedded in the (interior of the) given compact
manifold? This problem is considered as a variant of previous problems. We
present an affirmative answer via important theory in the singularity theory of
differentiable maps: lifting a given smooth map to an embedding or obtaining an
embedding such that the composition of this with the canonical projection is
the given map. Presenting this application of lifting smooth maps and related
fundamental propositions is also a main purpose of the present paper.

10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess