Immersions in the metastable range and spin structures on surfaces

Abstract

Recall that an immersion of a smooth manifold into another one is a map, which locally is a smooth embedding. Hence, smooth embeddings are immersions, but not vice versa. The basic classi¢cations of immersions and embeddings are inspired by the spaces of immersions and embeddings: Two immersions (embeddings) are regularly homotopic (diieotopic) if they can be connected by a path in the corresponding space. Smale and Hirsch [4] reduced the classi¢cation of immersions up to regular homotopy to homotopy theory. Hae£iger [3] did the same for embeddings up to diieotopy, in the metastable range. One may consider embeddings up to regular homotopy, but the diieotopy classi¢cation is more re¢ned. In this paper we consider immersions up to an equivalence, which is a generalization of diieotopy. It is called diieotopy equivalence. Two immersions are diieotopy equivalent if they diier by diieomorphisms, diieotopic to identity, of the source and target. We manage to classify generic immersions of a su⁄ciently highly connected manifold into Euclidean space, under the assumption that the dimensions are such that the dimension of the self intersection is 0, 1 or 2. The diieotopy equivalence classes are described in terms of the topological type of the self intersection and numerical invariants of additional structures (e.g. spin structure), which appear on the self intersection. This description depends periodically on the dimension of the source manifold. When the dimension of the self intersection is 0 or 1, the period is 2. When it is 2, the period is 4. The self intersection of a generic immersion in the metastable range is particularly simple. There, a generic immersion has no triple points and at a double point the intersection is transverse. This implies that the self inter