Immersions of surfaces in 3-manifolds

Abstract

THE problem of the classification of immersions of submanifolds of codimension at least one up to regular homotopy was reduced to a homotopy theoretic problem by the methods of Hirsch and Smale [6]. These techniques however are awkward to apply. James and Thomas [7] used the Hirsch-Smale results to enumerate the immersions of a surface into Iw3. Their technique was nonconstructive. In $1 we will give a classification of immersions of a surface, F, into a 3-manifold, M, in a given homotopy class, up to regular homotopy. It will be shown that there is a correspondence between H’ (F; Z,) and the regular homotopy classes homotopic to any given mapf, : F + M. This correspondence is l-to-l unless both F and M are non-orientable, in which case it may be l-to-2. The classification leads to an explicit construction ofall regular homotopy classes of maps within a given homotopy class of maps. We carry out this construction in $2. In 43 we study exactly what occurs during the process of (generic) regular homotopy. This allows us to define various invariants of the regular homotopy class of an immersion. One such invariant allows us to state precisely which regular homotopy classes can be realized by embeddings in Iw3. We solve the problem of when regular homotopy classes can be realized without triple points. Finally we obtain a new proof of a theorem of Banchoff on the number of triple points of immersed surfaces in Euclidean space. We would like to thank Larry Taylor and the referee for their helpful suggestions.