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Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle | | Sergiy Maksymenko
; Bohdan Feshchenko
; | | Date: |
25 Nov 2014 | | Abstract: | Let $f:M o mathbb{R}$ be a Morse function on a connected compact surface
$M$, and $mathcal{S}(f)$ and $mathcal{O}(f)$ be respectively the stabilizer
and the orbit of $f$ with respect to the right action of the group of
diffeomorphisms $mathcal{D}(M)$. In a series of papers the first author
described the homotopy types of connected components of $mathcal{S}(f)$ and
$mathcal{O}(f)$ for the cases when $M$ is either a $2$-disk or a cylinder or
$chi(M)<0$. Moreover, in two recent papers the authors considered special
classes of smooth functions on $2$-torus $T^2$ and shown that the computations
of $pi_1mathcal{O}(f)$ for those functions reduces to the cases of $2$-disk
and cylinder.
In the present paper we consider another class of Morse functions
$f:T^2 omathbb{R}$ whose KR-graphs have exactly one cycle and prove that for
every such function there exists a subsurface $Qsubset T^2$, diffeomorphic
with a cylinder, such that $pi_1mathcal{O}(f)$ is expressed via the
fundamental group $pi_1mathcal{O}(f|_{Q})$ of the restriction of $f$ to $Q$.
This result holds for a larger class of smooth functions $f:T^2 o
mathbb{R}$ having the following property: for every critical point $z$ of $f$
the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial
$mathbb{R}^2 o mathbb{R}$ without multiple factors. | | Source: | arXiv, 1411.6863 | | Services: | Forum | Review | PDF | Favorites | |
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