[en] 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 $T(e\stackrel{¯<!--\; ??\; -->}{f})$ of the immersion $e\stackrel{¯<!--\; ??\; -->}{f}:<!--\; :\; -->S^1\to <!--\; ???\; -->M_f\subset <!--\; ???\; -->{\mathbb{R}}^2$, where $\stackrel{¯<!--\; ??\; -->}{f}$ is the lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $⟨<!--\; ???\; -->[f]⟩<!--\; ???\; -->\subset <!--\; ???\; -->{\mathrm{\pi <!--\; ??\; -->}}_1(M)$.

Namely, immersions $f,g:<!--\; :\; -->S^1\to <!--\; ???\; -->M$ 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 $\mathrm{\nu <!--\; ??\; -->}(f)$ is nonorientable, then $S(f)=S(g)$; if $M\ne <!--\; ???\; -->S^2$, $M\ne <!--\; ???\; -->\mathbb{R}{P}^2$ and the bundles $\mathrm{\nu <!--\; ??\; -->}(f)$ and $\mathrm{\nu <!--\; ??\; -->}(g)$ have orientations $o$ and $o^{\prime }$ compatible with respect to the homotopy, then $T(e_o\stackrel{¯<!--\; ??\; -->}{f})=T(e_{o^{\prime }}\stackrel{¯<!--\; ??\; -->}{g})$, 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 $T(e_o\stackrel{¯<!--\; ??\; -->}{f})-<!--\; ???\; -->T(e_{o^{\prime }}\stackrel{¯<!--\; ??\; -->}{g})$ are reduced to the winding number of the lift of a certain null-homotopic immersion $f\mathrm{\#<!--\; \#\; -->}{g}^{\ast <!--\; ???\; -->}$ 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 $T(e_o\stackrel{¯<!--\; ??\; -->}{f})$ of the immersed circles. The proofs use the h-principle and are not very complicated.

Bibliography: 13 entries. (paper)