RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 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

Full text: PDF file (625 kB)
First page: PDF file
References: PDF file   HTML file

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

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
\Bibitem{Mel18} \by S.~A.~Melikhov \paper Immersions of the circle into a surface \jour Mat. Sb. \yr 2018 \vol 209 \issue 4 \pages 38--53 \mathnet{http://mi.mathnet.ru/msb8914} \crossref{https://doi.org/10.4213/sm8914} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3780078} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2018SbMat.209..503M} \elib{http://elibrary.ru/item.asp?id=32641399} \transl \jour Sb. Math. \yr 2018 \vol 209 \issue 4 \pages 503--518 \crossref{https://doi.org/10.1070/SM8914} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000436042300003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85049840907}