Invariants of Generic Immersions

An immersion of a smooth manifold M into a smooth manifold W is a smooth map with everywhere injective differential. Two immersions are regularly homotopic if they can be connected by a continuous 1-parameter family of immersions. An immersion is generic if all its self-intersections are transversal. In the space F of immersions M → W , generic immersions form an open dense subspace. Its complement is the discriminant hypersurface, Σ ⊂ F . Two generic immersions belong to the same path component of F − Σ if they can be connected by a regular homotopy, which at each instance is a generic immersion. We shall consider the classification of generic immersions up to regular homotopy through generic immersions. It is similar to the classification of embeddings up to diffeotopy (knot theory). In both cases, all topological properties of equivalent maps are the same. An invariant of generic immersions is a function on F−Σ which is locally constant. The value of such a function along a path in F jumps at intersections with Σ. Invariants may be classified according to the complexity of their jumps. The most basic invariants in this classification are called first order invariants (see Section 2). In [2], Arnold studies generic regular plane curves (i.e., generic immersions S1 → R2). He finds three first order invariants J+, J−, and St. In [4], the author considers the case S3 → R5. Two first order invariants J and L are found. In both these cases, the only self-intersections of generic immersions are transversal double points and in generic 1-parameter families there appear isolated instances of self-tangencies and triple points. The values of the invariants J± and J change at instances of self-tangency and remain