Immersions of the circle into a surface

Abstract

We classify immersions of a circle in a two-dimensional manifold in terms of elementary invariants: the parity of the number of double points of a self-transverse -approximation of , and the winding number of the immersion , where is the lift of to the cover of corresponding to the subgroup . Namely, immersions are regularly homotopic if and only if they are homotopic and the following additional condition is satisfied: if , or , or the normal bundle is nonorientable, then ; if , and the bundles and have orientations and compatible with respect to the homotopy, then , where is the standard embedding of the oriented surface (an annulus or a plane) in . In fact, for homotopic immersions and both numbers and are reduced to the winding number of the lift of a certain null-homotopic immersion to the universal covering of . The immersions 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 up to regular homotopy, in terms of the invariants and of the immersed circles. The proofs use the h-principle and are not very complicated. Bibliography: 13 entries.