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 $e\bar f:S^1\to M_f\subset\Bbb R^2$, where $\bar f$ is a lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $\left<[f]\right>\subset\pi_1(M)$. Namely, immersions $f,g:S^1\to M$ are regular homotopic if and only if they are homotopic, and if $M=S^2$ or $\Bbb R P^2$ or the normal bundle $\nu(f)$ is non-orientable, then $S(f)=S(g)$, whereas if $M\not= S^2,\Bbb R P^2$ and $\nu(f)$, $\nu(g)$ have orientations $o$, $o'$, compatible with respect to the homotopy, then $T(e_o\bar f)=T(e_{o'}\bar g)$, where $e_o$ is a standard embedding of the oriented surface $M_f$ (an annulus or a plane) in $\Bbb R^2$. In fact, for homotopic immersions $f$, $g$ both $S(f)-S(g)$ and $T(e_o\bar f)-T(e_{o'}\bar g)$ boil down to the turning number of a lift of a null-homotopic immersion $f\# g^*$ 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 $M\ne\Bbb R P^2$ 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 $M$.