Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Investigating additional structures on preimages of regular values of
  smooth maps induced from the manifolds via the Reeb spaces and applications   The Reeb space of a generic map is the space of all connected components of
preimages of the map. Reeb spaces are fundamental and useful tools in the
theory of Morse functions and higher dimensional variants and their
applications to geometry. These spaces are polyhedra compatible with natural
simplicial structures of the manifolds in considerable cases. For example, for
Morse functions, fold maps, and proper stable maps, which are useful and
fundamental tools in these studies, they are polyhedra whose dimensions are
equal to those of the manifolds of the targets.
  In the 2010s, Hiratuka and Saeki found that the top-dimensional homology
group of the Reeb space does not vanish for smooth maps such as proper stable
maps having some preimages containing a component which is not (oriented)
null-cobordant and which has no singular points. Recently the author found some
extended versions of this fact formulating and using cobordism-like groups of
equivalence classes of smooth and closed manifolds.
  The present paper concerns cases where manifolds and preimages with no
singular points of maps may have additional algebraic topological or
differential topological structures other than differentiable structures and
orientations. We show a theorem for this case. We also give some explicit
applications of the result where the preimages have spin or spin^c structures
induced from the manifolds of the domains and structures of framed manifolds
for example. As important applications, we present new explicit construction of
spin cobordisms and spin^c cobordisms for example.

23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-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
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc