Immersions of the circle into a surface

S. A. Melikhov

Sbornik. Mathematics, Tome 209 (2018) no. 4, pp. 503-518

Résumé

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.

Détail

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