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Mat. Sb., 2018, Volume 209, Number 4, Pages 38–53 (Mi msb8914)  

Immersions of the circle into a surface

S. A. Melikhov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: We classify immersions $f$ of a circle in a two-dimensional manifold $M$ in terms of elementary invariants: the parity $S(f)$ of the number of double points of a self-transverse $C^1$-approximation of $f$, and the winding number $T(e\overline f)$ of the immersion $e\overline f\colon S^1\to M_f\subset\mathbb R^2$, where $\overline f$ is the lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $\langle[f]\rangle\subset\pi_1(M)$.
Namely, immersions $f,g\colon S^1\to M$ are regularly homotopic if and only if they are homotopic and the following additional condition is satisfied: if $M=S^2$, or $M=\mathbb R P^2$, or the normal bundle $\nu(f)$ is nonorientable, then $S(f)=S(g)$; if $M\ne S^2$, $M\ne \mathbb R P^2$ and the bundles $\nu(f)$ and $\nu(g)$ have orientations $o$ and $o'$ compatible with respect to the homotopy, then $T (e_o\overline f)=T(e_{o'}\overline g)$, where $e_o$ is the standard embedding of the oriented surface $M_f$ (an annulus or a plane) in $\mathbb R^2$.
In fact, for homotopic immersions $f$ and $g$ both numbers $S(f)-S(g)$ and $T(e_o\overline f)-T(e_{o'}\overline g)$ are reduced to the winding number of the lift of a certain null-homotopic immersion $f#g^*$ to the universal covering of $M$.
The immersions $S^1\to M$ considered above can be smooth or topological; a smoothing theorem is proved showing that this difference is irrelevant. We also give a classification of immersions of a graph in $M$ up to regular homotopy, in terms of the invariants $S(f)$ and $T(e_o\overline f)$ of the immersed circles. The proofs use the h-principle and are not very complicated.
Bibliography: 13 entries.

Keywords: immersion, winding number, parity of the number of double points.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work was supported by the Russian Science Foundation under grant no. 14-50-00005.


DOI: https://doi.org/10.4213/sm8914

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English version:
Sbornik: Mathematics, 2018, 209:4, 503–518

Bibliographic databases:

UDC: 515.162.6+515.163.6+515.164.6
MSC: Primary 57N35, 57R42; Secondary 57R10
Received: 03.01.2017 and 04.09.2017

Citation: S. A. Melikhov, “Immersions of the circle into a surface”, Mat. Sb., 209:4 (2018), 38–53; Sb. Math., 209:4 (2018), 503–518

Citation in format AMSBIB
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