Immersions of the circle into a surface

Abstract

We classify immersions $f$ of $S^1$ in a $2$-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 turning number $T(e\bar{f})$ of the immersion $e\bar{f}:S^1\to M_f\subset {\mathbb{R}}^2$, where $\bar{f}$ is a lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $⟨[f]⟩\subset {\pi }_1(M)$. Namely, immersions $f,g:S^1\to M$ are regular homotopic if and only if they are homotopic, and if $M=S^2$ or $\mathbb{R}{P}^2$ or the normal bundle $\nu (f)$ is non-orientable, then $S(f)=S(g)$, whereas if $M\ne S^2,\mathbb{R}{P}^2$ and $\nu (f)$, $\nu (g)$ have orientations $o$, $o^{\prime }$, compatible with respect to the homotopy, then $T(e_o\bar{f})=T(e_{o^{\prime }}\bar{g})$, where $e_o$ is a standard embedding of the oriented surface $M_f$ (an annulus or a plane) in ${\mathbb{R}}^2$. In fact, for homotopic immersions $f$, $g$ both $S(f)-S(g)$ and $T(e_o\bar{f})-T(e_{o^{\prime }}\bar{g})$ boil down to the turning number of a lift of a null-homotopic immersion $f\#g^{\ast }$ to the universal cover of $M$. Here "immersions" $S^1\to M$ are either smooth or topological; we include a smoothing theorem, which shows that there is no difference. We also classify immersions of a graph in $M$ up to regular homotopy in terms of the invariants $S(f)$ and $T(e_o\bar{f})$ of immersed $S^1$'s. The proofs are based on the h-principle. The point of this unsophisticated note is to simplify [10] and [11], where a classification of immersions of a graph in $M$ was obtained for $M\ne \mathbb{R}{P}^2$ in terms of a rather laboriously defined "winding number" of a pair of homotopic immersions $S^1\to M$ (rather than of an individual immersion) with respect to a given vector field with zeroes on $M$.

In Russian. 17 pages; v4: added an introduction

References in corpus (2)

• The compression theorem III: applications
• Whitney's formulas for curves on surfaces