Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Special framed Morse functions on surfaces   Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse
functions on $M$, and $\mathbb{F}^1$ the space of framed Morse functions, both
endowed with $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed
Morse functions is defined. We prove that the inclusion mapping
$\mathbb{F}^0\hookrightarrow\mathbb{F}^1$ is a homotopy equivalence. In the
case when at least $\chi(M)+1$ critical points of each function of $F$ are
labeled, homotopy equivalences $\mathbb{\widetilde K}\sim\widetilde{\cal M}$
and $F\sim\mathbb{F}^0\sim{\mathscr D}^0\times\mathbb{\widetilde K}$ are
proved, where $\mathbb{\widetilde K}$ is the complex of framed Morse functions,
$\widetilde{\cal M}\approx\mathbb{F}^1/{\mathscr D}^0$ is the universal moduli
space of framed Morse functions, ${\mathscr D}^0$ is the group of
self-diffeomorphisms of $M$ homotopic to the identity.

13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
16
The categories of this paper are: stat
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat